So conics is what we are going to begin with today, but let me give you an overview of this big topic called conics. And what we are going to cover under this big topic conics. So for you in coordinate geometry, you have primarily two things. One is your straight line chapter and the other is conics. Conics itself is made up of five chapters. So there are almost five chapters that we are going to touch upon in conics. And this forms the heart and soul of your coordinate geometry. Let me tell you, conics is something which many people find it difficult in comparative exams. The main reason for that is it is covered in a very, very, you can say superficial level from your CVSC point of view or from your NCRT point of view. But the type of questions that you get in the competitive exams are to a considerable depth. So the NCRT or the CVSC curriculum is not able to match the requirement of the conics. Now you will say, sir, this will happen in almost all the chapters. Yes, it happens in almost all the chapters, but this gap is much wider in conics. I'll tell you the reason why. If you look at your NCRT textbook, conic section is covered within 2325 odd pages, maximum 30. I have not exactly counted it. But if you talk about conic section from your J point of view, let's say if you open up a standard J book, any standard J book, you realize that they have either dedicated a full book to this conic section or they have at least 300 to 400 pages worth of conic section content. So there is a wide gap. Okay. So here is the deal. First of all, I'm going to cover as of now from your school exam point of view. For the J point of view, we need much more time, which I don't think so we'll be able to get because most of your final exams are scheduled for February end. So if you see, we have January and that too, like, you know, last five, six days will be going into your new year time or Christmas time. That's our vacation time. And, you know, January full will be getting and maybe few weeks of February. So we will not be able to do justice to a conic section from J point of view. Just to give you an idea, Circus, the very first chapter of conics itself takes around 15 hours. That means almost four classes will go in just doing circles. Okay. That means one month will be only gone for circles. Similarly, parabola, ellipse hyperbola pair of straight lines, they're also going to take at least three, three classes each. So we don't have that much of time. And before that your school semester exams are also going to start. So we will do it. Don't worry. I'm not saying we will not be doing conic sections to that depth. We will be doing it immediately after your after your school semester exams are over. So there is a gap between your 11th and 12th. In that gap, we are going to go full swing with respect to conic sections. So what are there in conic sections or in conics? The first topic that we are going to do and we are going to start with this topic today is circles. A sweet and simple chapter. But many of the learnings that you will have from circles, you will be taking it forward to other codecs. Okay. So there are certain aspects that you're going to learn in circles, which you're going to use as it is, as it is for parabola, ellipse hyperbola, etc. Okay. The second chapter that we are going to talk about is parabola. Okay. Or parabolas, what do we call it? Okay. So circles parabola. We'll be talking about ellipses. We'll be talking about ellipses. We'll be talking about hyperbola. Okay. And finally, we'll be talking about pair of straight lines. By the way, this part is not for your CVSC curricula. It's not there in your NCRD textbook. Okay. Not for your school level. Okay. So in your school, if you see, they would not be doing the chapter pair of straight lines. But yes, it is an important concept in coordinate geometry, which will be taking up immediately after we are done with circles, parabola, ellipse and hyperbola. Okay. So this is our today's agenda circles. Okay, let's talk about it. Now, many people first asked me, sir, why do we call them as conics? Okay. So these are all 2D figures. Mind you, none of them is a three dimensional figure. But it is basically obtained by a meeting of two 3D figures. So let's talk about why the name conics was given to this. So let's let's take a simple structure over here, which we call as right circular double code. Okay. This is called a right circular double code. Okay. It looks like an R glass or kind of a thing. Okay. So this looks like an R glass. Okay. This is a right circular double code. Okay. Why this called a right circular double code? Because the normal to the base. Okay. Is basically your axis of the ellipse, or sorry, axis of the code. So if you see, this is a double code. Okay, this is a double code. Okay, as you can see, there are two cones combining at this point. And the axis makes a 90 degree with the base of the code. Okay. So that is why it is called right circular double code. Let me write it down. Right circular double code. Okay. Right circular double code. These two are called the naps of the code. These two are called the naps of this right circular double code. Okay. These red lines that you see, they are called the generators. These are called the generators. See, by the way, there is nothing to do with naps and generators in the chapter per se, but it is good that you should know it. It's a good information to have. Okay. Now this right circular double code when it interacts with a plane, a 2D plane, everybody knows a plane, right. So plane is a two dimensional structure. In fact, it's a three dimensional structure, but it doesn't have a third dimension. We normally make it in a 3D structure. So when it cuts this right circular double cone, let's say I make a slice. Let's say I make a slice like this. Yeah. Okay. Okay. And this slice will create a cross section over here. This cross section will be a corner. And in this case, if you see, this is going to become a circle. If you see from the top. Okay. So please put your eyes on the top. Okay. This is your, let's say eyes. You are watching it from the top. It is going to look like a circle. Right. So circle is a corner, which is formed when a right circular double cone is in the section. The circular double cone is intersected by a plane in such a way that the plane is making 90 degrees with the axis. Okay. So let's say I call this angle as beta angle. Let me just take this angle as beta angle. Okay. By the way, all of you would be aware of this fact that this is called the semi vertical angle. So please note this down. Alpha is called the semi vertical angle, semi vertical angle. Okay. Now, if the angle beta made by this plane, if the angle beta made by this plane and the axis is 90 degrees, then the monic obtained will be that of a circle. Okay. Then a circle is going to be obtained. Okay. I'll not write is formed. I just write circle is meant to be obtained. Okay. So you can very well imagine that there is a right circular double cone like this.  And after what has been chopped off, you remove and see from the top. Okay. If you see from the top, you'll see still see a circle, isn't it? Okay. So conic, which is a circle is formed by when your beta angle is 90 degrees. Now, let me ask you this question. What if your beta angle is lesser than, let me write it in yellow just to make it consistent. What if the beta angle is lesser than 90 degrees? Okay. But greater than alpha. Then what will happen? Can anybody tell me, can anybody imagine that scenario? Can anybody imagine that scenario? Okay. So let's say, let's say I make another diagram. Let me make another diagram. How is the axis defined? Axis is basically a line which is, you know, passing through the vertex of this. This is called the vertex point, vertex point. And in this right circular double cone, this is a line which is perpendicular to the two bases of the board. That is the axis again. Okay. Let me make a right circular double cone once again. Okay. So let me just make the basis here. Now, this time your plane is cutting this one of the naps in this fashion. And if you see the cross section that will be generated would be like this. Okay. This is the cross section. Again, you're seeing it from this angle. So in this case, what is happening is that the angle made by the plane, this angle, if you see this angle, this angle will be lesser than 90 degree, but it would be more than, it would be more than alpha. Agreed? And in such cases, the conic that you are going to obtain will be that of an ellipse.  Okay. So it would be something like this. I'll just make a quick diagram of an ellipse. Okay. We are going to take it very soon. This is how an ellipse will look like. Circle you already know. Okay. This is circle. This is ellipse. Is it fine? So ellipse is also obtained by interaction of a plane with a right circular double cone when it interacts with one of the naps of the cone at such an angle which is lesser than 90 but greater than alpha. Clear? How an ellipse is formed? Any questions? Okay. Now let's look into the third case. In our third case, our plane is going to cut the naps in this way. Let me show you how it is going to cut the naps. I'm just making one instance here. There you go. So it has been sliced in such a way that this plane is parallel to the generator. This plane is parallel to the generator. In other words, the angle of the semi-vertical angle and this angle are going to be the same. That means if your beta becomes equal to alpha, in that case the structure generated will be that of a parabola. Parabola is basically a name given to the path taken by a body which is thrown under gravity.  That ball is going to take a path and that path is called a parabola. That is a parabolic path. The trajectory will be that of a parabola. I'm sure you would have done it in projectile motion in physics also. Yes, ellipse is kind of oval. Yes, ellipse is oval structure. Now, how does a hyperbola get formed? Let us try to look into this. A hyperbola is formed when you're, let me again draw one. I should have taken a snapshot of this right circular double cone. It takes a little bit of time to draw it again and again. But anyways, this is the last demonstration of calling. So if your right circular double cone is cut by a plane in such a way that, let me just make a plane here. So if you see the cross section that will be generated would be something like this. I'm just making a rough sketch. This will be the cross section. I didn't show the cross section here. One minute. I have to show the cross section here also. So this is also a cross section. So this cross section will be that of a parabola. So parabola who, people who have already done it in school, they already have seen the structure of a parabola. It basically looks like this. Okay, this is a parabolic structure. The headlamps of your car, they are parabolic in nature. Have you ever seen the side view of your car lamps? I'm sure some of you who own at least this model Corolla model. I remember Corolla model. You can see the projector lamps on the side. It looks like a parabolic. It's a parabolic reflection. So hyperbola is obtained when the angle made. So let me just show it to you. When this angle, the angle that it makes with the axis is either zero degree or at least should be lesser than alpha. So in such cases, your angle would be either zero degree or lesser than alpha. So when it is either zero degree or lesser than alpha or somewhere between them, it is going to cut both the naps. And because of that, you will see these two cross sections appearing and that together is called a hyperbola. That together is called a hyperbola. So hyperbola structure looks like this. Now many people say, sir, I feel hyperbola is just double the two branches of a parabola. See, it looks very similar to it, but there are a lot of differences that you will see when we actually study these topics in detail. And now, yes, can you show parabola? This is a parabola. I will show you all on GeoGibra. Will that be fine, Karthik? I'll show all the diagrams on GeoGibra. And now if this plane is made to pass through the center of, or you can say the vertex of this right circular double-core, something like this. Okay, so I am basically making a plane which is passing through the vertex of this right circular double-core and you see this cone from this angle. Okay, if you watch it from this angle, you would see that there would be two lines like this getting formed. Okay, I am just showing you a side view. I at least should pass through this thing. Okay, so these two lines would be that of a, these two lines would be that of a, let me write it here. So these two would be pair of straight lines. So pair of straight line is actually called a degenerate conic because it doesn't cut the generators. It actually passes through the, through the vertex, the meeting point of the two naps. Okay, so pair of straight lines is also called a degenerate conic. Okay, so it is also called a degenerate conic. See, these are all informations which is good to have. It is not going to impact your, you know, topic understanding. Okay, maybe in school they don't talk about this, but always a question will be running in your mind that how it is called a conic section. Okay, that conic figure, you want to see a cone. That's what you're trying to say.  Let me just show you. Conic diagrams. Okay, since we are already, yeah, I think this is a beautiful figure to show you what is happening. This one there. Let me just take this one. I don't know if it takes to some site or something. Let's take a snapshot of this. Yeah, so I'll just put the image over here. And just to show you how it rhymes with our understanding. Okay, so see how a circle has been formed. So a circle has been formed when this plane, this blue plane cuts this right circular double cone at an angle, which is perpendicular to the axis. This is what we have basically shown you. So here your beta is equal to 90 degrees. Ellipse is formed when your beta is between 90 degree and alpha. Okay, between 90 degree and alpha. Okay, ellipse, sorry, parabola is formed when your beta is equal to alpha. Okay, and a hyperbola is formed when your beta is somewhere between zero to alpha, including zero, zero is fine. Okay, I should mark beta. Okay, so this this angle is your beta. Is this fine here? This angle is your beta. Okay, and here this angle is your beta. And here the plane just draw a line, just draw a line, which is basically passing to that on the plane. Okay, this is your beta. Okay, beta between clear now setu makes sense. Yeah, yeah, yeah. So on that plane, you basically draw a line like this. Okay, think as if you're looking at a plane from side view. Okay, that line, what angle will it make? That is your beta plane. When you have a side view of the plane, it will look like a line only. That angle is beta. Okay, so this is a plane. No, let's say, so let's say if you are looking from such an angle where it appears just like a line. Are you getting my what and this is your axis. So this angle is your beta. Okay, by the way, it is looking like a line, but it is the side view of a plane. See, mobile phone, let's say, this is the mobile phone. It's a 2D figure. If I look at it from here, I will see a line like this. Okay, what that line makes. So for your, your view, it will look like this. This is your plane. Okay, it is looking like a line to you. This is axis. Okay, the angle between them is your beta. Correct. Clear everybody. Okay, so this is just for your basic understanding why these two dimension figures are actually called phonic sections. Okay, so let's start with our chapter itself, which is your circles circles. I'm not going to cover everything. I'm just going to cover up concepts which are going to be helpful for your school exams. So, Karthik, won't the plane be one of the generators? No, the plane will be parallel to the generator is actually a line which rotates to generate a right circular double code generator is not a plane by the way generator was a lot. If you do a cross section that cross section shows you two lines like this that two lines are called generators. If you see from the front, if you slice the right circular double code, you see the two lines like this those are called generators. Now, somebody I think Karthik, you wanted to see these images on Georgie. I'll show you that. Give me a second. So, circles you have already seen. Okay, let me draw one for you so that everybody sees it again. I'm sure all of you have seen a circle before. Okay, this is a circle. This is a ellipse.  You must be wondering how is the writing all these equations. This is an ellipse. Okay, let me show you a parabola. This is a parabola. Okay, and let me show you a hyperbola as well. Okay, this is a hyperbola. So, let me just switch one on at a time. So, this is circle. This is ellipse. This is a parabola. Now, the way to pronounce it, many people in India pronounce it very incorrectly, parabola. It's not parabola. It's parabola. Parabola. Everybody please pronounce it once. I may not be able to hear you, but yes, just pronounce it. Parabola. Yeah, pronounce it. Not parabola and all. Okay, and this is your hyperbola. There are some differences. There are many differences between parabola and hyperbola, which probably from the diagram, you will not be able to appreciate, but I will show you when we do the chapter. Is it fine? Any questions? All right, so let's start with our circles chapter. So, for every conic, we are going to learn a locus definition. See, even though all these conics have come from your interaction of a plane with a right circular double O, but these all conics actually satisfy locus conditions. Okay, so they're all 2D figures. They're all two dimension figures, right? You can show them on a 2D figure, on a 2D plane. So, they are all following some kind of a locus definition. So, if that locus definition is met, you will be able to get one of the other conics of it. For example, in circles, the locus definition is well known to everybody. What is the locus definition of a circle? Anybody can tell me? What is it? It's a locus of a point. Okay, it's a path. It's a path traced by a point moving in a plane in such a way that the distance of the point from a fixed point is a constant. Okay, so let me revisit this definition. First of all, it's a path traced by a point moving in a plane. Please remember that it cannot move in a three-dimensional structure, else it will become a sphere. A different sphere is a three-dimensional object. We'll talk about sphere in our class 12. Such that its distance from a fixed point, that fixed point is called the center of the circle. You already know it. There's not a news to you. You already know it. It's a constant and this constant value is called the radius of the circle. Okay, everybody knows it. Now, the same locus definition is going to help us in order to find the equation of a circle. See, guys, let me tell you one thing. Many people ask me this question. Sir, circles we have done now in class 10. Why we need to do it again? See, in class 10, the circle that you did was from geometry point of view, where you talked about different theorems related to a circle, different laws related to a circle. There was no equation written for any kind of a circle. Now we are doing analytical geometry or what we call as a coordinate geometry where we'll be talking with respect to equations. Okay, so we are now learning the same topic with respect to our equations. So we need to write down the equation of a circle based on this locus definition. So what is that equation? Very good, Prism. Prism has already written it. So let us say this is our fixed point C and let's call it as alpha beta, alpha beta fixed value and let's say this point P is a moving point. Okay, so this is a moving point. Okay, but it moves in such a way that this distance is always a constant value or fixed value, which is let's say R. Okay, so the path traced by this particular point under such locus condition would be that of a circle. As you can see, anywhere it goes, let's say it comes here, it is still going to be R, it comes here, it is still going to be R and so on and so forth. Right, so using this locus definition, we will be writing this condition that PC will always be R. What is PC? PC will be under root of H minus alpha whole square K minus beta whole square equal to R. Okay, just square both the sides. Just square both the sides. When you square both the sides, you end up getting H minus alpha whole square K minus beta whole square is equal to R square. And after we have written this, we can generalize this. Right, how do you generalize it by replacing your H with an X and K with a Y. So when I do that, I will end up getting the equation like this X minus alpha the whole square Y minus beta the whole square equal to R square. Please note this down. This equation, we have called it as the center radius form of the equation of a circle. So this equation is given a special name. What is that special name center radius form of the equation of a circle. Please note it down. Is it clear? Any questions with respect to the derivation part? Is it fine? Okay, in this equation, if you take your center to be origin, right? So if you make your center, please note that this is an important point here. So if your center is taken to be origin, okay, if alpha beta becomes your origin. Okay, then the equation becomes X square plus Y square is equal to R square. Okay, this form of the equation is called the standard form of the equation of a circle. Please be very, very careful about the use of these names which I'm giving to these equations. What is center radius form? What is standard form? It should be very clear to you. So center radius form is where you have been. See, you can say standard form is a special case of a center radius form. But when I use the word standard form, you should always realize that the origin is going to be your center. Okay, your center is going to be origin. Now people ask me this question. So why do we call this as a standard form? See, many a times you will realize that we will be deriving a lot of properties based on these standard forms. Yes, so for standard form, origin is the center. Okay, so these are called standard forms because these standard forms are like the frog in the laboratory. Right, we do a lot of experiments on frogs, lizards and all right. In the same way standard forms would be our you can say experimenting tool for us to derive or prove many properties. Okay, so that is why they are called standard forms. Is it fine? Any question with respect to what is the center radius form and what is the standard form of a circle? Any questions you let me know. All set? Okay, let's do some analysis. Let's do some analysis of this equation, the center radius form equation. So I will just write it down again x minus alpha the whole square y minus beta the whole square equal to r squared. Okay, if you expand it, if you expand it, this is what you're going to see x square minus two alpha x plus alpha square y square minus two beta x plus beta squared and just take this r squared to the left. Okay, so this becomes x square plus y square minus two alpha x minus two beta y. Okay, I'm so sorry I wrote x here, it should be y my dear. Okay. Yeah, plus alpha square plus beta square minus r square equal to zero. Okay. Now this equation, if you just write in a sober fashion without you know, see this is a constant right so you can call it as a constant let's say I call it as C. This x square let me let it be x square this y square let it be y square. And just to give it a proper shape I'll be calling it as two gx. Okay, and just to you know give this a proper shape I'll be calling it as two f y. So if you see this type of an equation is what you will be getting for almost all the circles. I should not use the word almost. In fact for all the circles. You can get such an equation. This equation is what we call as the general form of the equation of a circle. This is called the general form. So we had a, we had a center radius form. We had a standard form under that. And there is now a general form. Are you getting my point.  See, it's actually a convention that we follow any book that you pick up, whether it is an NCRT book or whether it is a J level book or even a foreign authors book, they will always write the general form like this. Remember there was a general form for a straight lines equation also. A x plus b y plus c equal to 0. Same way, even for a circle, this is called the general form. And it is always recommended and advised that you should leave your final answer in the general form unless until asked to do otherwise. Okay, in school also when you're solving the question, many people leave the answer in the center radius form. That's not a good practice actually you should leave it in your general form like this. Okay, so some comparisons which you can directly draw from this is that your alpha is actually minus g right if you compare this and this you'll get alpha is minus g. Similarly, if you compare these two you'll get beta as minus f. Okay, in other words, if the equation of a circle is provided in the general form, you can claim its center to be at minus g minus f. Okay, this is to be noted. Okay. Okay, one thing sorry I was supposed to write that down. Many times in the equation of a circle, some we can say some smart question setter may multiply with let's say some constant let's say you can multiply it with a 5 or a 10. Okay, whatever you multiply is it. If you multiply this entire equation with some number. Please note that under that situation, you cannot, you cannot use this, you know, formula for your center. So many times when I write the general form, I categorically mentioned this to the students that in the general form, the coefficient of x square and y square should be made one. Okay, if it is not a one that means if it is multiplied purposely by the question setter by some other non, you can say non unity number, that's a 5, 600 whatever he multiplies it with, then that particular form will not be the general form. And if it is not a general form, you cannot use this as the coordinates for the center I'm repeating this again. I'm repeating this again. Right, many of you would initially make mistakes, no corresponding to this so I'm specifically warning you please be careful that don't use this as your center coordinates, if the form given to you or if the equation given to you is not the general form. Makes sense. Is it clear? Don't worry, we'll take examples. And if somebody's making an error, we can, you know, try to rectify it through those examples. Second thing that you can compare here is that your C value is actually going to be G square. Sorry, alpha square. Yeah, if you just compare this with this, your alpha square plus beta square minus R square is actually C value right. So from here, let us try to make R the subject of the formula. So R can be written as or R square can be written as alpha square plus beta square minus C. Correct. Yes or no. And just now we saw that alpha is minus G and beta is minus F. Okay. In other words, your R square becomes G square plus F square minus C. That is to say that your R value is under root of G square plus F square minus C. Okay. So another formula, please add this to your formula list here as of now, that if somebody gives you the equation in the general form, this is the radius expression or this is the radius value that you should be obtaining from it. So these two should be remembered. I'm just making a bubble around here. So this is to be noted and remembered. Is it fine? Any questions? Any concerns? Okay. Now, we add something very interesting over here. Radius is always greater than equal to zero for a real circle. Okay. For a real circle. So a real circle means a circle which you can see on the Cartesian coordinates. Its radius should always be greater than equal to zero. If your radius is less than zero, we call it as an imaginary circle. So these are not under our scope right now. We will be not talking about imaginary circle and all. Okay. If radius is equal to zero, many people ask me, sir, won't the circle reduce to a point? Yes. In that case, it becomes a point circle. Okay. So a special case under this, if your R becomes equal to zero, we call it as a point circle. Okay. Then it is called a, then the circle is called a point circle. Circle is called a point. Is it fine? Imaginary circles are those circles whose radius is negative. Okay. So we will not be discussing about those kind of a circles, but mathematics has a unique way of justifying that there could be imaginary circles of that kind. Okay. Imaginary circles cannot be seen. They are not real. They cannot be realized. Okay. There's no doubt. I mean, no questions are going to come based on that. Of course. Is it fine? Any questions, any concerns you have so far with respect to the general form? Let's take a small question on it. But before that, please ensure you have copied everything down. So I'll repeat once again, the summary of this page, whatever we discussed in this page, if you have been given the general form of the equation of a circuit. Now, what is the general form? General form is a form exactly like this in a general form. The coefficient of X square and Y square is always one. Okay. So once you know it's a general form, this formula could be used to get the center and this could be used to get the radius of the circuit. Let's take a question. Can I go to the next slide? Everybody. Those who have copied. Let's go to the, let's go to the next slide. So if they asked to give general form up to reduce the coefficient of X square and Y. Yes. Yes. If they asked you to write down the equation in the general form, you need to reduce the coefficient of X square and Y square to one. Absolutely correct. Let's take questions. Here you go. First question. Find the center and radius of the circle. I would request everybody to leave your response on the chat box. So give me the center and give me the radius of the circle. Yes. Anybody. Very good. What about the radius? Many of you are forgetting to write the radius. So write both the aspects and center and radius and then press and enter so that I get both your answers at one place. Very good. Siddharth. Arundati. Nice. Okay. Vashna. Vashna, are you sure? Just check you're working once again. Very good. Okay. All right. So let's discuss it out. So let's bring first of all, all the terms to the left. Something like this. Okay. Now ask, just answer one question of mine. Is this in the general form? Is this in the general form? Is this in a general form? You'll say, no, why? Because the coefficient of x square and y square are two here. Okay. Right. One important thing I would like to highlight over here. Even before we start solving this problem. If you go to the previous slide. Okay. There is one analysis that you need to do over here. If you look at this equation. The general form equation, let me write it over here. Okay. Some important points to be noted. Important points to be noted. In any circle, if you see the coefficient of x square and y square would always be equal to each other. Okay. This is true for any circle. See, in this case, they are one one, but even if you multiply, let's say, I decide to multiply this entire equation with let's say 100. Even then the coefficient of x square and y square will be equal to each other. Okay. That is point number one. So it is a very good identification for us that if in a clinic, the coefficient of x square and y square are different. That clinic will never be a circuit that clinic can never be a circle. So sometimes question like this is also asked. So they will give you an equation second degree equation. And they will ask you, does this represent a clinic? Does this represent a circle? So the first thing you should check is my coefficient of x square and y square equal to each other. If that itself is not true, please note that it is a necessary condition, but not a sufficient condition. So what are the different between a necessary condition and a sufficient condition? Sufficient condition means if that condition is met, your job is over. But in our case, it is a necessary condition. That means if that is not satisfied, it is not going to be a circle. But if it is satisfied, it may not be a circle. Try to understand this fact. Are you getting my point? Let's say if you don't have a fever, right? You need not be healthy. But if you have a fever, you are definitely not healthy, right? Because there is something wrong. That's why you have got a fever, isn't it? Okay. Good example, right? Good medical example. So if the coefficient of x square and y square are not equal, then it will definitely be not a circle. But even if they are equal, it need not be a circle. Are you getting my point? That is the meaning of necessary condition. So in every circle, the coefficient of x square and y square will always be equal. So in our example also, if you see, they have given 2-2. Okay. So let me just go back again. Yeah. So as you can see here in our question, our coefficients are both equal, 2-2. So it is a circle for sure. Next thing that I would like you to note down. Second thing, in a circle, there is no x-y term, right? A circle equation will never have an x-y term. Now, let me tell you, this is also a necessary condition. Are you getting my point? This is also a necessary condition. But are they together sufficient? The answer to that also is no. That means even if the coefficients of x square and y square are equal, and even if there is no x-y term, it doesn't mean it is going to be a circle. Right? Okay. So what makes, what is the third condition which will make it sufficient? For that, you have to wait for some time till we start our straight lines topic. Okay. So as of now, please note down that these two are the necessary conditions, but not the sufficient conditions. Necessary condition for a circle. Conditions for a circle. Okay. To represent a circle. So if you see there is an x-y term sitting in any second-degree equation. For example, somebody gives you like this. He'll say x square plus y square minus xy plus 6x plus 4y plus 2 equal to 0. Is this a circle? What will you say? Is this a circle? What will you say? He'll say, no, this is not a circle. It cannot be a circle. Why? Because there is the x-y term sitting here. It cannot be a circle. Circle should not have any x-y term. Getting my point. Okay. Let me give you another example. X square plus 2y square plus let's say 5x plus 4y plus 3 equal to 0. Can it be a circle? Can it be a circle? No. Not a circle. What is the reason? You immediately point out, sir, see the coefficient here is 2. The coefficient here is 1. They are not equal. Right. Because of that, it is not a circle. Are you getting my point? Okay. But if I write something like this. Okay. I found this on the web for. Yo. Sorry. The CD got activated. Yeah. So this is a case. This is a case where it is not going to represent.  Are you getting my point? Okay. There are many cases that you'll see later on that even if the coefficient of x square and y square are equal. Even if the coefficient of x square and y square are equal, and there is no x y term, still it will not represent a circle. Still it will not represent a circle. Okay. Now many of you would be thinking, can I give some example of that nature? Let's take, let's take this example. Yeah. Let's take this example. I mean, I'm just going to, I'm just going to, I'm just going to, I'm just going to, I'm just going to, I'm just going to, let's take this example. I mean, I'm just, I'm just trying to create one example over here. So let's, let's try to see what is meant to come out from here. So let's say I have an example like this. Square and you'll have something like this. X plus Y. I wanted to get the coefficient of XY canceled. Okay. Let me just, let me just see some example present. Maybe I can, what example can I give you? Give me some time. I'll just think of some example where X square term and Y square term coefficients are same. There's no XY term, but it cannot, but it'll not represent a circle. Okay. We'll take that example in some white and sometimes just give me some time. I'll think of it and I'll get back. Okay. Okay. We're all coming back to our problem. Sorry. We didn't solve this problem yet. Getting examples very quickly is not, you know, easy thing. You'll have to, you know, satisfy all the conditions and write one. Yes. So in this case, if you see, this is not a general form. The reason being this coefficients are not equal to one. Okay. So what we are going to do here is we are going to divide by two first of all. So when you're going to divide by two, you end up getting something like this. So once you have done it, you can now compare this with two G, compare this with two G and you compare this with two F. Okay. In other words, your G becomes negative three by four and your F becomes five by four. And remember when I told you the general form of a circle, the center was at, where was the center? Center was at minus G minus F. So for our case, it will be three by four comma minus five by four. Yes or no. Yes or no. Correct. Okay. And what used to be the radius for such case? Radius for such case used to be under root of G square plus F square minus C. A Karthik C, two G is minus three by two. So what is G? Sorry. Three by two. What is G? Minus three by four. No. Got it. Two F is five by two. So what is F? Five by four only. No. Okay. Yeah. So radius would be under root of G square. G square will be nine by 16 plus F square, which will be 25 by 16 minus C minus C. Minus C means minus of minus seven by two. So how much does it give you? Let's try to simplify this. This is 34 by 16. This is going to be plus seven by two. Seven by two is as good as 56 by 16, which is as good as under root of 90 by 16, which is three root 10 upon four. Okay. So this becomes your radius of this given circuit. If you don't take that, you will get it wrong. It is not an option to not take it. Getting the point. Is this fine? Any questions? Okay. So such questions can be asked as a one market in your semester exam or maybe UT's. By the way, December 6th. That means Monday. Was your seniors semester one. Board paper of maths. Okay. Just ask them how was the paper? Because if you hear from me, you'll say, I use her is trying to scare us. He is trying to make us work hard. See that anybody should do. But let me tell you the paper was lengthy. The paper was lengthy and only those students, especially our center students have all done well. Those students who were fine tuned to prepare, preparing for these competitive exams under time bounded manner. They were only able to complete the paper and they only did well. Right. So please understand this that now the CVC board exams, even those are not that simple. Right. You can't just ask your board exams also with casual, you know, preparation. You have to be very, very meticulous in a time-mounted manner. You have to start solving problems. Karthik, she are done. If you're done, do let me know. Karthik done. Okay. Let's take a question on your standard form. Sorry, on your center radius form. Find the equation of a circle concentric with the circle and passing through minus two, minus seven. Everybody knows the meaning of concentric, right? Having the same center. Done. So this question is basically indirectly giving you the coordinates, sorry, the center of the circle. And it is giving you a point on the circle so that you are able to figure out its radius. Okay. So center is concentric with the center of some other circle. I'm just taking, I'm just making a rough diagram. Please don't be, you know, this is not an exact scenario. Okay. So this circle center is given to us. Okay. So our requirement is we need this circle. What is the equation? We need this. And it is also given that the center is the circle contains minus two, minus seven. Okay. So let's figure it out. Very good. I am getting responses. So everywhere. See. So this center can be obtained by using our general form. So minus G minus F used to be the center. Okay. So let's take all the C. So minus G. So see here, this is two G. This is two G. So two G is minus eight. So minus G will be four. Okay. This is two F. This is two F. So minus F will be minus three. So this is your center of our required circle. And what is the radius? Radius is nothing but the length CP. Okay. So what are the distance of four comma minus three from this? Let's try to figure it out under root of four minus minus two, the whole square and minus three minus minus seven, the whole square. So that makes it under root of 36 plus 16 root 52 root 52. So if I write down the center radius form, this is the answer that I should be getting X minus four, the whole square Y plus three, the whole square equal to root 52 square. Okay. Let's simplify it. X square minus eight X plus 16 Y square plus six Y plus nine equal to 52. Let's try to simplify it. It becomes X square plus Y square minus eight X plus six Y minus 27 equal to zero. Okay. This becomes your answer. But now let me tell you this is not a smart way to solve the question. This is not a smart way to solve the question. You should never solve this question in this way. Okay. Because it involves unnecessary finding of the center. You can solve this question even without finding the center. How? Let me show you. See, you would all recall that when we did straight lines. If you're given one line and you were asked to write a line parallel to that line. You remember AX plus BY thing, you know, we used to keep it the same and only the constant term we used to change and just find out the constant term and put it back. Do you recall that way of solving the problem when we had to find the equation of a line parallel to a given line in the same way, think as if this yellow circle is a parallel thing to this white circle. In other words, if you want to find the equation of a yellow circle, everything is going to be the same. Let me show you the second method. Everything is going to be the same means X square, Y square minus AX minus AX plus six Y till here it will be the same only constant term will be different. Because the constant term is dependent on the radius, but these terms, they are not dependent on the radius. They only depend on the center. And if the center is not changing, even these terms are not going to change. Are you getting a point? So how do you find the C value? Just simple. Make this circle be satisfied by minus two minus seven. So minus two minus seven satisfies this equation. So that will give you four plus I'm just putting the values four plus 49 plus 16 minus 42 plus C equal to zero. Let's solve for C here.  Just put this back and your job is done. Your equation of the circle is obtained. Nothing else. Is it fine? So these are the tricks that you need to keep in mind. While solving the question. Right rational. See how easy and time efficient it was. Getting my point. Clear everybody? Any questions, any concerns with this approach? Okay. So many people ask me this question. Sometimes I casually say the same concept of parallelism that we have between lines in circle that concept becomes con concentric circles. So the concentric circles could be treated like a parallel circles. Okay. That's a synonymous. You can say idea in case of a circle. All right. So moving on to the next form. So we already did center radius form. In fact, a special form of center radius form was your standard form. Now we're going to talk about the diametric form of a circle. So what is the diametric form? Now this is a situation based form. See everything, the main concept, the underlying principle is the same across all the forms. Right. So in center radius form, if somebody gave you the center and the radius, then you use the center radius form. Correct. But if somebody provides me the diametrical opposite ends of a circle, let's say somebody provides me, you'll say, Hey, I've given you these two end points of A and B or these two points A and B, X1, Y1 and X2, Y2. Can you get the equation of the circle? Can you get the equation of the circle? How will you find that out? Okay. So one thing you will say, sir, I will find the center first. Correct. I will find the center. I will find the radius and I will write down the, I will write down the center radius form. Correct. But let me tell you that would involve finding the center using midpoint formula, finding the radius by using the distance formula and then you constructing a center radius form and then you're simplifying it to the general form. Wow. That will, that will involve at least a bit of your time. Right. So we have all those, you know, you know, processes, instead we can use a locus definition here as well. See how locus becomes so important for us, isn't it? So can I say this circle will be locus of all such points, H, K such that the extreme points A and B subtend the right angle at B, isn't it? Down, down. Correct. Yes or no? Can I say circle is this, this circle is the locus of all such points with subtend or on which a right angle is being subtended by this AB. If AB happens to be the diameter. Yes or no? What about A and B themselves? Yes. They form the limiting case of the scenario. Okay. So let's try to get our equation of a circle by the use of this idea itself. So can I say slope of MP and slope of BP will be, you know, the product will be minus one. Yes or no? Yes or no? So what is the slope of AP? Slope of AP will be K minus X, sorry, K minus Y1 upon H minus X1. What is the slope of BP? K minus Y2 upon H minus X2. This is equal to minus one. Let's try to simplify it a bit. Okay. Don't worry. You don't have to do it every time. I'm just doing it once in for all so that when we get the result, we can keep using it again and again. So let's bring it to the other side. It becomes H minus X1, H minus X2 plus K minus Y1, K minus Y2 equal to zero. Now let's generalize it like how we always do in finding the locus.  So please make a note of it. This is what we call as the diametrical form of the equation of a circle. So in this form, you don't need to find the center. You don't need to find the radius. You don't need to write the center radius form. You don't need to simplify it either. Is it fine? Any questions? I'm not able to read the name of the person who has sent it. Who has asked this last question? Karthik Sanoj. Karthik, why your name is not fully visible to me? Once again, I just add some issue with the system. Okay. Karthik, why does minus one? Karthik, what is the product of two lines? What is the product of the slopes of two lines which are perpendicular? So you got your answer? It happens. Many times, obvious things don't strike. Okay. Anyways, let's take questions. Let's take questions. Let's take this question. The side of a square is made by these lines, X equal to two, X equal to three. Okay. Y equal to one, Y equal to two. Okay. Just to make them on a coordinate axis. So this is your X equal to two, let's say. Okay. This is X equal to three. Let's say this is Y equal to one line and this is Y equal to two line. Okay. As you can see, this builds up a square in the middle here. Okay. So we have to make a circle on the diagonals of the square. So this is how the circle will look like. Yes or no? Right? So let's say this is the diagonal. Okay. There are two diagonals here. You can use any one of the diagonals. Okay. We need to know what is the equation of this circle? Equation is what we desire. Done. If you're done, let me know your result on the chat box. Done. Excellent. Excellent. Okay. Let's discuss it out. See, this line is X equal to two line. This line is X equal to three line. This line is Y equal to one line and this line is Y equal to two line. Let us call these, call the square as ABCB. Okay. Let's write down the coordinates of any two, any of the diagonals. So there are two diagonals you can see on your screen right now. Let's write down the coordinates of AC, A and C. So A will be two comma one and C will be three comma two. Okay. So this is good enough for us to write down the equation of a circle.  Okay. Y minus one, Y minus two equal to zero. Is it fine? Any questions?  Just write it in a general form. This is going to be the general form of the circle. Now many people will ask me, sir, if I have used BD as my diagonal, would I have got a different answer? No. Let's use that also. If you use BD as a diagonal, remember B is three comma one and D is two comma two. Okay. So if you would have used, this is, this is with AC as diagonal. So AC as a diameter. Okay. Let me use, let me use BD as the diameter. And let's see, am I getting a different answer altogether? Or is it going to give me the same figure? Ideally it should give me the same figures. So if BD is used as a diameter, you'll get X minus X one, X minus X two. As you can see, both the, both these factors are coming here also. So they should not be any difference ideally. Plus Y minus Y one into Y minus Y two. As you can see, again, same figures are coming here ideally. So there would be no difference. There would be no difference. You'll still get the same answer. X square plus Y square minus five X minus three Y plus eight equal to zero. Okay. Same answer as whatever you got. Is it fine? Any questions? Any concerns here? Do let me know. Is it fine? Any questions? Okay. Let's try one more. Find the equation of the circle. The end points of whose diameter are the centers of these two circles. If you're done, you can just write it done as well. No need to type it out. If you feel it's going to take a lot of time to type. A done would do. Okay. Vaishnav. Adya is done. Very good. Done. Okay. So very simple. So let's say, let's say these are the two circles. I'm just drawing smaller, small, small circles. Okay. And through the center of these two circles as diameter, you are making another circle. Let's make one in yellow. Okay. Something like this. Yeah. So with the centers. I'm sorry. With the. Extimate centers of these two circles as a diameter. You have made another circle, which is this yellow one. I need to know its equation. Yeah. So first of all, what are the centers of these two circles? Let's write them down. So A will be, if I'm not mistaken, A will be minus three comma seven. B will be two comma minus five. I hope all of you are now familiar how to find out the center quickly. Okay. So use the diameter form to get the equation of this yellow circle.  Y minus Y two equal to zero. Simple.  And I think the constants would be minus six coming from here and minus 35 coming from here. Okay. That's minus 41 equal to zero. Am I right? Makes sense. Makes sense. Clear everybody. Okay. So we have also now seen the diametrical form. Let us now talk about parametric form. That's another form coming your way. Parametric form. Now, first of all, what is a parametric form? Is there a fixed parametric form for a circle, et cetera, et cetera. A lot of questions normally are asked with respect to this topic. So let me take this up one by one. So first of all, what is a parametric form? First of all, there is no one parametric form for a curve. There can be numerous parametric forms of equations you can write for a curve. Let me begin with a simple example, then we'll come back to circle. Let us start from a line. If I say there's a simple line X plus Y equal to one. Okay. So this form of the equation is called the Cartesian form. What is Cartesian form? A form where you are directly relating your Cartesian variables X and Y. By relation like this, that is called a Cartesian form. Okay. So Cartesian form is a form where you are directly relating the variables with each other. Correct. So the same equation, if I write it like this, X equal to T and Y equal to one minus T, where T is a parameter, where T is a parameter. Do you realize that if you just add them, they will still give you a one, isn't it? So such equation is what we call as the parametric form. Okay. But a very big question arises, sir, is this the only parametric form that could be that I could write for this particular line? No, there could be several parametric forms. For example, if you can write X as one plus T and Y is equal to let's say T, you can say minus T. Okay. That also works. So even if you add them, it'll give you one only, right? Yes or no? Parametric form is not sacrosanct. Parametric form is not... Parametric form, I have not completed yet carpet. I'm just beginning to understand the concept. Parametric form is where you are trying to relate two variables to the third parameter, to the third party. See, I'll give you a simple example. Villages. I don't know how many of you have been to villages. Ruler word, rural India. Okay. There they will not say husband wife. They will say he's the father of Chintu. She's the mother of Chintu. I am giving an example, isn't it? So they automatically become husband and wife. Are you getting my point? So X and Y think as if they're husband-wise. So instead of saying they're married and they're husband-wise, you're saying X is the father of this guy, or this child, and Y is the mother of that child. So automatically they become husband-wise. Yes or no? So in case of a parametric form, you are trying to use a similar analogic. So you are trying to call X and Y by different parameters. Yeah. So here instead of directly writing X and Y relation with each other, you are connecting them through a third party, which is called the parameter. Now many of you would be asking, sir, is parameter a variable? Now let me tell you parameter is a constant which can vary. Like you said, this is an oxymoron. It's a constant which can vary. How is that possible? See, I'll give you another example. All of you have an Aadhaar card number, right? Correct. Setu, you have an Aadhaar card number. Yes or no? Correct. Is that Aadhaar card number fixed for you, or does it keep changing every day? Fixed. Correct. Karthik Saloj, do you have an Aadhaar card number? Yes. Is it fixed for you? Karthik says yes. Is Setu's Aadhaar card number and your Aadhaar card number different? Yes. Now see, here is the connect. So everybody has a fixed Aadhaar card number, but it varies from person to person. So Aadhaar card number becomes a parameter. Are you getting my point? So T can keep changing for different, different points. So T is like that Aadhaar card number for a X comma Y. So think like X comma Y is a resident of a curve. Curve is a country. So let's say this is a curve.  So X comma Y is a resident of that country. This is your country. Take as if this is your country. And this is one of the resident of your country, of that country. So this resident will have an Aadhaar card number T. It will have a parameter T. So this T is fixed for that guy. X comma Y. So if you change your X comma Y, let's say if your X comma Y is now some different point. Let's say I call it as X1 Y1. So for this X1 Y1, it is T1. And if I now go to X2 Y2, it will become some other number. Okay. The parameter will change to T2. But it is going to be fixed for X2 Y2. It is not going to change for X2 Y2. So every point has a parameter associated to it. I am getting my point. Every point lying on the curve has a parameter associated to it. Okay. So that parameter will be fixed for that point. But if you change the point itself, the other point will have a different parameter. Again, think as if this is Setu and this is Aadhaar card number. This is Karthik. This is Aadhaar card number. Right. Are you getting my point here? So for any curve, you can actually suggest a parametric form. Okay. Like this. Right. So it could be numerous parametric forms could be written for it. Now many people will ask me this question. So why do we need a parametric form? I'm happy with Cartesian form. So the use of parametric form is in facilitating us to choose a point on that curve. Now let's take this example itself. If this is your straight line and I'm asking you on this straight line, choose a point. How would you choose a point? Let me ask somebody. Let's let me ask. Adya, how would you choose a point on this line? How would you choose a point on this line? Let's say choose any point on this line. How would you choose it? You'll say, sir, alpha one, alpha beta or X1, Y1, correct? Yes or no? But if you choose a point like this, let me tell you it has got some issues with it. What is the issue? First of all, two unknowns have to be chosen. Means you're increasing the number of unknowns. Please note more unknowns, more problems. So while solving questions, try to keep the count of the number of unknowns as less as possible. Second thing is you have to always carry this burden on your head that X1 plus Y1 will be equal to one because it satisfies this curve. Yes or no? So why to do all this thing when I can use my parametric form and say, hey, if I have to choose a point on this curve, I can choose it as T comma one minus T. Overdone. So what is the advantage here? Only one unknown T. So if I know T, I know the point. Correct. And you never have to take the curve into your consideration.  Now you understood what is the logic behind use of parametric form. And let me tell you, even though it is not taken in your CVSC curriculum, most of the JEE problems when it comes to coordinate geometry. Many of the JEE problems, if you, I mean, of course, the problem should be oriented in that way. If you use your parametric forms, you can solve it in a much easier and faster way. Okay. So in one, you know, straight, you know, you can say summarizing it in one straight line, you can say parametric forms facilitates us to choose a point on that curve with less number of unknowns. Okay. Not for a constant for a given curve, it's constant for a given point on that curve for every point has a parameter for it. Example, I showed you know, see, this is a country, it has got several points on it. So think as if those points are like residents of that country. Got it. So every point will have a parameter for it. So, for example, if I have to choose a point here, I would choose it like this, as you mean T is the parameter for that. Right. If I have to choose another point, let's say, so I'll choose something like, let me call it as T1 so that you can understand. Okay, so let me call it as one T21 minus T2. If I have to choose another point, I'll call it as T31 minus T3. Are you getting my one. So, see, instead of choosing six unknowns for three points, you would use six unknowns x1, y1, x2, y2, x3, y3. You only need three unknowns. The number of unknowns are drastically reduced. Got the point. So two things should be clear as of now. What is a parametric form? Is it a sacrosanct figure or it can change as per the person who is, you know, using it or person who wants to use it. And what is the meaning of a parameter? These two things should be clear to you. So please understand, parametric form is not a fixed form. That means if I ask what is the parametric form for this, five different people can give five different answers and they can all be right at the same time. Are you getting my point? But Cartesian form is a fixed form. You can't have, if I ask you, can you give me another Cartesian form for this equation? There is no other Cartesian form. Cartesian form is only one form. But you could have several parametric forms like this. There's no end to it. Somebody can write 3 minus t plus 2 plus t. So 3 minus t comma 2 plus t or x equal to 3 minus t, y is equal to, sorry, t minus 2. That can also become a parametric form. So so many answers can be there. So I can give you another answer like let's say x be 3 plus t and y is equal to minus 2 minus minus t minus 2. Even if this you added, you'll get one only. So this can be another form. So point here is there is no sacrosanct. There is no fixed parametric form for a curve. It can change depending upon the person who is trying to use it. Okay. But many times you will see that there has been a, you say conventional, you can say a parametric form which has been followed since long. So we'll be talking about those conventional forms also while I'm dealing with this particular concept. So let's go to our circles now. Let us start with, again, I will name the topic as parametric form. Let's say I give you a circle. Let's say I start with a standard form of a circle. Okay. x squared plus y squared is equal to r squared. This is a Cartesian form. Can I, can you suggest me a parametric form for this? Let me ask you this question instead of me telling you the answer directly. Can you suggest me a parametric form for this? Anybody? Can I not write this situation like x is equal to r cos t, y equal to r sin t, where t is a parameter. Correct? So if you eliminate the parameter, automatically this form will lead to the Cartesian form. Correct? So if you just do x square plus y square, what do you end up getting? You will end up getting something like this. Okay. Take r square common. Correct? It ends up giving you, it ends up giving you this. Yes or no? Correct? Now is this the only, you know, parametric form? You say no, sir.  Correct? This can still give you the same Cartesian form. No difference. Correct? Yes or no? I can give you one more form. In fact, many more can be formed, but I'll give you one more. r by root 2 cos t sin t and r by root 2 cos t minus r by root 2 sin t. I can write it like this also. Right? But ask yourself, if you are choosing a point on this circle, would you prefer choosing it as r cos t comma r sin t? Or will you prefer choosing it like this? Which, which will be more preferable? Will you prefer choosing it like this? Or will you prefer choosing it like this? What will you prefer? The first one, right? Yes or no? Because this looks so complicated. This is complicated. It's complex. Okay. Let me write complicated else you would think complex number. Okay. So, but this is still correct. I'm not saying this is a wrong parametric form. It is still correct. Okay.  Okay. How to write a parametric form. Please note that there is nothing like the parametric form. Some questions, some books, I have seen they use this word, write down the parametric form. They cannot be the parametric form. They could be a, so it's a, it's, you could suggest a parametric form for that curve because it is not a sacrosanct. It is not fixed. Right. Isn't it? So it could be written in several forms. So let's take some more practice on this. Let's say if I ask you for this circle, for this circle. Okay. Suggest a parametric form for the above circle. Okay. How will you write it? See, you just have to use a parameter and write your X and Y independently related to that parameter. So that if I eliminate that parameter, I should get this equation back. Okay. So I'll make your life simple. Had I given you something like this, capital X square, capital Y square is equal to R square or equal to five square. How would you have written it? Nikhil, absolutely right. Okay. But Nikhil, try to understand here. When you write that your Y will always be positive as per your answer. But is it a mandate? Think, think, think. Can't my ordinate be negative for some point? So your, your parametric form is just showing half the circle, top part. Okay. Now see it. And another thing to be asked yourself, if you're choosing a point on the circle, will you choose it like under root of 25 minus T square plus three? That doesn't look so good. I don't like it. Okay. At least I don't like it. Maybe somebody else would like it. See how would I choose it? From the previous example, you could write X as R cos T, Y as R sin T. And just do one more step here. Change your capital X with X minus two, change your capital Y with Y minus three and bring this two and three to the other side. And there you go. It becomes X equal to two plus five cos T and three plus five sin T. That's it. This is a parametric form. This can be an answer. Okay. So here there is no restriction like my X is negative or my Y is always positive or something like that. Is it fine? Any questions? Okay. So with this example, we can also generalize the situation. That means so in general, if somebody gives you the equation of a circle like this, okay, and he asks you to suggest a parametric form, then you can always write X as alpha plus R cos T and Y is equal to beta plus R sin T. Sin cos positions can also be interchanged. It will not make it wrong anyhow. Okay. So this could be a parametric form for this curve. Is it fine? Any questions? Any concerns? Please get this idea clear because it's going to help you in your higher versions of coordinate geometry, which we are going to take immediately after your semester exams are over. Can we take another question? Let's take first of all this question. This is an easier question to solve. They are asking you for the Cartesian form, for the two curves over here, which are mentioned in parametric form. So let's do the A1 first. We'll come to B1 after some time. So if your parametric form for a given circle is written like this, of course theta is a parameter. They have not mentioned it. What is the Cartesian form? This is a parametric form. They're asking you what is the Cartesian form? Okay. So our agenda here is to eliminate the parameter. Right. So what do you do here? So I'm sure everybody would have done this exercise in class 10 stignometry. Right. So eliminate theta from these two. How will you eliminate? You'll see it's simple. X plus four by five is cos theta and y plus three by five is sin theta. And everybody knows the Pythagorean identity cos square theta plus sin square theta is one. Correct. So if you square them and add them, you get X plus, okay, and this square equal to one. In other words, X plus four whole square, y plus three whole square is equal to 25. This is going to be your Cartesian. Sorry, parametric Cartesian form for this equation. Yes, sir. Which most of you have got it. Well done. Please do the B part as well. B part. Correct. So let's do the B part also. I'll just rewrite the question here. So we had a cos alpha plus B sin alpha.  Okay. And here, even if this is not mentioned, alpha is a parameter. Okay. A, B are fixed values. A, B are not parameters. Guys, don't get confused between parameters and fixed values. A, B are fixed. They're fixed numbers. Let's say two and three. Alpha is a parameter which will keep changing for every point on that curve. Okay. So what are the way to eliminate alpha? I am sure you would have done this exercise earlier as well. You just square them and add them. Let's see what happens because of that. Let's say if I do this. So if you square this, it will become a square cos square alpha. This will become something like this. Okay. And the other term will become a square sin square alpha B square cos square alpha and minus two AB sin alpha cos alpha. This will be your X square plus Y square. Now these two will anyways get canceled. This two will become a square. This will become B square. So you'll end up writing. You'll end up getting a square plus B square on the arches. Yes. Yes or no. So this becomes your answer.  He made a such a ugly parametric form for this. If I were there, I would have made this parametric form cos alpha. This is much easier to look and deal with. Okay. So this is another answer to the same question. Let's say if you wanted to have another parametric form for this. Okay. This could be used as another parametric form. So that is what I was trying to say. Parametric forms are not sacrosanct for a curve. A curve can have 100 parametric forms. Some of them may be very complicated. Some may be very easy to, you know, look at. Maybe very easy to apply. Use also are you getting my point? What I'm trying to say. Okay. So let's take another one. Any questions? Any concerns? Please do highlight. Let's take find the parametric form of this circle. Okay. Again, I would like to correct this question. This is not a right way to frame the question. Okay. Sorry. From whichever book this was taken the question. The author has not written it in the right way. They should say suggest a parametric form. That is the right word to use. Suggest a parametric form for this curve. See, it's like, you know, our hard card number is only one identification you have, but you could also be known by your passport number. Correct. You could also be known by your, you know, birth certificate. I don't know whether you carry pan or not. People are also known by their pan card number, pan card number, especially when it comes to income tax related issues. Correct. So it could be like, you know, I can say, give me an identification or, you know, give me one of your identification. So it is not a fixed identification that you can carry. You can carry other card also passport. Also you can give me, you can give me a driving license. Also if you carry one can give me a pan number. Also if you carry one. Same way. So when you are writing a parametric form, it is like somebody is asking you for identification of a point on that curve. Getting my point. If you're, if you're done, you can just say I'm done. Okay. No need to type it out. Just tell me and yes. So basically, you know, if you, if you write it like this, let me write the center radius form for this. Oh, sorry. It is X plus P by two, Y plus P by two, the whole square. And I think this is going to be P square by four P square four. So P square by two will come. Okay. In other words, you can write it something like this. Okay. Let me rewrite this. So here if you try to treat it like X minus alpha whole square, Y minus beta whole square. Equal to R square. Remember what was the parametric form we wrote for such cases. Alpha plus R cost T beta plus R scientists. This was one of the parametric forms which you could use. Similarly, can I use. Can I use X equal to. Alpha alpha is minus P by two. Plus R R is P by root two. Cost T and Y is minus P by two plus P by root two scientist. Okay. This is fine. So this could be a suggested parametric form. Getting it. Any questions? Any questions, any concerns? If the constant is yes, if there is no constant term in the equation of any curve that curve will pass through origin, whether it is a circle, whether it is a straight line, whether it is any curve, any curve in general. Okay. All right. So I think we can take one more question. Then we'll take a break on the other side of the break. We will basically discuss few typical problems which are asked on circles in school exam as well as competitive exam. So this is a question which says find the equation of the following in Cartesian form. Again, it's a simple to convert. And if the circle, if the curve is a circle, find its center and the radius. Okay. Let's, let's see whether it is a circle. And if it is, what is the center and the radius. So please do this question, everybody. Very good. Anybody else? Okay. Sure. Okay. Let's discuss it. So alpha is a parameter here. I want to eliminate alpha. So while eliminating alpha, let us say, do we stumble upon a circle equation? Let's check. So in order to eliminate alpha, I will bring one to the other side. Okay. Square it. I'll bring three to the other side here. Square it. And I'll end up getting four on the other side. Which is clearly the equation of a circle. And this is clearly a circle equation. Right. And what is the center of the circle? Very easy. Center of the circle will be minus one comma three. And radius of the circle will be two. Right. Correct. The ship. Correct. Siddharth Karthik also correct. Everybody's correct. Okay. So here we'll take a small break. On the other side of the break, we are going to take some different varieties of questions asked on circles. For example, if you give me three points on the circle, how do you find the equation? If you give me any two point and the center lying on a line, how do you find the equation? So there are different types of questions which are being asked. We'll take those varieties of questions after the break. Right now the time as per my watch is six or two. Let's meet exactly at six, 17 PM after a 15 minutes break. Okay. On the other side of the break will take up those concepts. Okay. See you in 15 minutes time. Okay. Enjoy your break. Now we are going to run through some problems which are normally asked in the school and comparative exams where you would be given certain kind of an information. And based on that information, you have to find the equation of a circle. So up till now, whatever we have done, for example, if the center and the radius is given to us, we use the center radius form. Okay. If there's somebody has given me diametrically opposite ends, we use the diametrical form. Okay. So the same way you would be given certain information by which you can directly or indirectly find the radius and the center of the circle and write down any type of equation that you want to. Let us take an example here. Let's take an example here. Let's say I want to write down the equation of a circle which passes through these two points and has its center on four X plus Y equal to 16 line. Okay. So I wish to write down the equation of a circle which passes through these two points. Let me name them as A and B. Okay. And its center lies on its center lies on a given line. Let's say the line is like this. Okay. I'm just taking a how would I find the equation of such a circle? Now I would first give you one minute to think about it. Okay. And then we'll discuss it. Maybe, you know, you can take your time to solve it also and give me your final answer. Give me a final verdict on the chat box. First of all, how many methods are there to solve these kind of questions? See here. The first method that we can use here, let me just replicate the graph actually, the photograph actually. Yeah, let me just take a snapshot of this, not graph and diagram. Sorry, diagram. Okay. See one method to solve this would be one method to solve solve this would be you find the equation of the perpendicular bisector of AB. Okay. And we all know that the perpendicular bisector of the chord will be passing through the center of the circle. Correct. So you find the midpoint. You already know the slope of AB. So you would know the slope of the perpendicular to AB as well. Use a slow point form to get the equation of this line. Correct. Okay. So let's say this is line L1. This is line L2. So L1 and L2 will meet on the center. Okay. So here L1 and L2, they will meet at the center of the circle. So once you know center of the circle, you can find any of the distances from A or B. So AC or BC, they will give you the radius of the circle. Okay. So once you know the radius, once you know the center, you can write the center radius. Nikhil will come to your answer in some time. This is one way of solving it. Another way of solving it is let me pull the diagram once again. The second way of solving this question is you could assume the center to be some point. Now, here I would request you to tell me any or suggest me any parametric form for this line itself. 4x plus y equal to 16. Can you suggest me a parametric form for this line? So 4x plus y equal to 16. I could suggest I could write a parametric form for this as x equal to t and y equal to 16 minus 40. Correct. Yes. So I could take the center to be t comma 16 minus 40. Correct. Now use this distance that C A, let me write it down here. So use the fact that C A equal to C B and you get your t. Okay. So once you get your t, that means you know the C coordinate. That means you know the center then. And once you know the center, you can find the radius by using either the length C A or C B that will give you the radius. So again, write down the center radius form. Done. Is it fine? No, Siddharth, slightly mistaken. If you take y as t, x will become 16 minus t by 4, not 16 minus 40. So a small mistake in the way you have written it. Okay. Now there's a third approach also. And that third approach is what I am going to take to solve this question. So what I normally do, I'll say, okay, let the circle of the circle be, let the circle be x square plus y square plus 2 g x plus 2 f y plus c equal to zero. So I have assumed the general form of the equation of a circle. Okay. Now this equation must be satisfied by, this is satisfied by the points. 4 comma 1. I am sure I hope I have called it. Yeah. 4 comma 1 and 6 comma 5. So it is satisfied by 4 comma 1 and 6 comma 5. So if you put 4 comma 1 here, you get 4 square, 1 square, 2 g 4, 2 f 1 plus c equal to zero, which basically simplifies to 17 plus 8 g plus 2 f plus c equal to zero. Let's call it as 1. And it is also satisfied by 6 comma 5. So that will give you 6 square, 5 square, 2 g 6, 2 f 5 plus c equal to zero. On simplification, this will give you 36 plus 25, 61 plus 12 g plus 10 f plus c equal to zero. That's your second equation. Okay. Third equation is the center satisfies this line, 4x plus y equal to 16. Okay. So minus g minus f, which is your center, that must satisfy that given line. So that will give us minus 4g minus f is equal to 16. Let's call it as the third one. Okay. Now I have to solve for g, f and c from these three equations. So first let me do 2 minus 1. 2 minus 1 if you do, you will end up getting 4g. Okay. You will end up getting 8f. Okay. And cc will get cancelled off. And by the way, this will give you 44. So minus 44 will come here. Am I right? Let's call this as the fourth equation. Okay. Now let's add third and fourth. That gives us 7f is equal to minus 28. So f is equal to minus 4. Correct. Any problem so far. Take your time. I'm just waiting for, you know, 30 seconds. Just take your time. See what has been done. So I formed these three equations. One, two, three. Right. So from two minus one, I got the fourth equation. I added third and fourth and I got f value. So far so good. Any questions? So if I've got my f value, I can get my g value from 30 equations. So let us put it here. So 16. Oh, I'm so sorry. Not 16 minus 4g plus four is equal to 16. So minus 4g is 12. So g is negative three. G is negative three. Okay. So if we have got f and g, let us find out C as well by using my first equation. So 17 plus 8g. 8g will give you eight into minus three. Okay. 2f. 2f will give you minus eight. Okay. Plus c equal to zero. So what does C value come out to be 24 C value comes out to be if I'm not mistaken 15. So once you have got f, g and C, what are we waiting for? We'll have to just put it in the equation here. So our answer will become, I'll just write it on top, x square plus y square. 2g will become minus 6x. 2f will become minus 8y plus 15 equal to zero. This is going to be your answer. Is this fine? Any questions? Any concerns? Any questions? Any concerns? Absolutely right, Nikhil. I think your answer is matching with mine. Now many people ask me, out of all the three methods, what made you choose method number three to solve the question? See, in method number one and two, I found that I have to spend some time finding the center and then find the radius and then write down the center radius form and then again convert it to general form. So instead of why not just find f, g and C and automatically getting the general form. So why we need to find center separately? Why need to find radius separately? Why need to write the radius center, center radius form and then convert everything back to general form? So I feel it is my personal call. I mean, you can take any approach that you want to. It's absolutely your call. So I feel this approach is shorter of the two, shorter of the three I should say. It depends upon you how convenient you are in using these. So three methods I have told you, you're free to use any one of the three. Is it fine? Any question anybody has? Any question anybody has? Please do let me know. If you want me to scroll the screen somewhere, let me know. Okay. All right. So this is one type of problem that you can get in circle. Let's take another type. Oh, sorry. I think I opened them wrong. Okay. Let's take this one. This is another important type of question that comes in school exams also. Find the equation of a circle passing through three non-colonial points. Three non-colonial points. See, most of you would have studied this theorem in class 10 that there can, there is a unique circle that can be made to pass through three non-colonial points. Right. So the problem is now asking you to get the equation of a circle, which is passing through three non-colonial points. So let's say if I have any three non-colonial points, I'll just make like this. So I can always make a circle pass through them and this will be a unique circle. In the sense that they can only be one such circle, which can pass through, which can pass through these non-colonial points. Okay. I'm just trying to fit one. Yeah. Now it is. Yeah. So how will you solve this question? How will you solve this question? Again, there are several approaches. Okay. Approach number one. But before that, let me just take a snapshot of this. Maybe I would need this diagram once again. Approach number one. You could assume that the center is some alpha beta. Okay. Assume that this is alpha beta. Okay. So find, let me use B for this. Okay. So find your center P here. Okay. By using the fact that AP is equal to BP and AP is also equal to CP. Okay. So two equations, two equations, two unknowns you can solve for the center. So once you know your P, one center is known. A radius is nothing but any one of the distances PA. So once you have found the center and the radius, you could go for your center radius form that will give you the answer. Correct. This is approach number one. Approach number two is, let me just call the figure once again. Yeah. Approach number two could be you write down the perpendicular bisectors of AB and BC. Okay. So they will meet at the center of the circle. Correct. So you can find P through that. Yes or no. So the perpendicular bisectors of the chord will meet at P. So find P. Okay. By, by finding the, by finding the intersection of intersection of perpendicular bisectors. Of AB and BC. So once you found that out, your radius is nothing but your radius is nothing but either PA or PB, whatever, or PC, all of them are same. So find the center radius for that. But I'm not going to use the third approach, which is very much similar to what we discussed in the previous page. And before that, I would request you to solve this question so that we can match our answer. So can't you find the center using distance formula center using distance formula? How, how can you throw some light on it? If these three points are given to you, how the center, you know, do you know the center? All right. That's what I discussed. No center alpha beta. That's what you're talking about. Approach number one that I already discussed. Yes. So the third approach is where you're assuming that let the circle be x square plus y square plus two gx plus two f five plus c equal to zero. Okay. Now, of course, we know that these three points are going to satisfy the equation of the circle. Okay. So what are the three points one comma one. So one comma one. Two comma minus one, if I'm not mistaken. Yeah. And three comma two. Okay. Satisfy this equation. Okay. So if they do, then just put the value of x y one one. So two g one. Okay. So this gives us. See, I'm going to just simplify and write it going forward because this is not a rocket science. You can all do it at your own and also. So let's say I call this as one. If you put two and minus one, it'll give you two square plus one square five plus four g minus two f plus equal to zero. Let's call it as two. Okay. When you put three into you get nine plus four 13. Six g plus four f plus c equal to zero. Let's call it as 30 question. Okay. So how do I solve for many people say, sir, three equations. Oh my God, sir. How will I solve it? I feel difficulty in solving. See, there's no difficulty at all. All you need to do is pair by pair. You just start taking their difference. So take the difference of one and two. So do do two minus one. So if you do two minus one, you'll end up getting three plus two g minus four f equal to zero. Okay. Let's call it as the fourth equation. Then do three minus two. So you end up getting 12. Oh, sorry, not eight. Eight plus two g plus six f equal to zero. Let's call it as 50 equation. Let's subtract these two as well for minus five. So two minus five will give you minus five minus 10 f equal to zero. So f is minus half. Done. F is minus half. Okay. Let's check whether your answer was correct or not. So now put this f value in let's say the fifth one. Okay. Put it in fifth. Eight plus two g six f will give you minus three equal to zero. So g value is minus five by two. Okay. Put G and f in the first equation. So I think G was this is minus five f was minus half C. So C value comes out to be if I'm not mistaken for. So once you've got f G C, let's put it in the equation. So my final answer will become x square plus y square to G x will be minus x. I'm sorry, minus five x to f y will become minus y and C is plus four. There you go. This is the answer. Is this fine? Any questions? Any concerns with respect to this? Okay. See, I'm using this method that doesn't mean, okay, you will also use this every time it is up to your convenience. How was, how much was G? G was minus five by two. Is it fine? Any questions? All right. Let's take more questions. Okay. Let's take this question. The equation of a circle which passes through one comma zero zero comma one and has its radius as small as possible. I'm putting the poll on in case you're done, you can put your response on the poll. It's actually a super easy question. You just have to think before you start writing. Imagine a situation where there's a circle passing through these two points and its radius has been made as small as it can be. Okay, so you can very well imagine it. What is going to happen? Oh, wonderful. Somebody has already given a response and that too well within 30 seconds. Awesome. Good. Right. Yes. Yes. Absolutely. You're right. Anybody else? In fact, I can just see five of you responded on the poll. Okay. Should we discuss it now? Five, four, three, two, one. Okay. Just 11 of you responded and I got a very confused response. You can see BCND almost, almost equal words. So that shows confusion. Right. All right. So let's discuss this. Okay. Let me just make a actual diagram of the scenario. One comma zero. Let's say I assume it here. Okay. Zero comma one. I assume it here. Okay. You're passing a circle. You're passing a circle through these two points. Okay. Now in order to make a circle, which is having the smallest possible radius. Okay. Let me just drop some cases for you. Let's say this, these are, this is a circle, which is passing through this point. Do you think this is the smallest radius? No. It could be reduced. It could be reduced in this way where you start shifting the circle like this. Till you realize that the two points here actually become farthest apart. Okay. And that will happen when these two points have actually become your diametrically opposite ends. So when this becomes your diameter, in that case, your circle radius would be the smallest. Correct. So this was an indirect question where they're asking you to use the diametrical form. Correct. So it'll be X minus zero X minus one Y minus zero Y minus one equal to zero. So X square plus Y square minus X minus Y equal to zero option number T is right. Isn't it? Common sense. See, these are the indirect questions. They will not say directly that, hey, this is the endpoints of the radius. Sorry, endpoints of the diameter. Okay. So these are some indirect questions which are, which could be framed, you know, and asked given to you to solve. Okay. Let's take this one. Find the equation. Okay. Yeah. Find the equation of the circle which touches the axes and whose center lies on the line X minus two Y equal to three. Okay. I'm just making a small scenario out of this question. Oh, sorry. Okay. It's a scenario like this. Okay. Sidharth. No, no, not necessarily. It is not a necessary condition. What you're saying is not necessary. What if it is like touching and the center lies on this line? Does it have to be Y equal to X line? No, it can be any line. So Sidharth has given one response, but other than him, I think nobody has solved it yet. I'm waiting for a few more responses. Okay. Okay, Nikhil. Good. Anybody else? Okay. Let's discuss it. Only two people have responded to this question. Was it so difficult? I don't think so. See, first of all, what are the take away from the fact that the circle touches both the coordinate axes? Let's say, let's say I call the center as minus G minus S. Okay. What are the take away from the fact that this circle touches both the coordinate axes? Tell, tell, tell, tell, tell, tell past. Can I say this will be true? Right. Because this length and this length should be equal. So this length is what? This length is mod of minus F. This length is mod of minus G. They should be equal. Yes or no. Yes or no. Correct. Now this means G could be equal to F or G could be equal to minus F. Yes or no. Okay. So let's take this condition. First of all. Now, second thing that I can write it. Let me call this as situation number one. Let me call this as situation number two. Now, the other condition that is given to me is that the center should lie on X minus two Y equal to three. In other words, minus G plus two F equal to three. This is my second, you can say second equation. So this is my first equation. This is my second equation. Okay. Now let's take situation number one and second equation. So if G is equal to F and minus G plus two F equal to three, let's see what do we get? Let's see what do we get?  That means F becomes three. And since G and F are equal, it means G is equal to three as well. Okay. And not only that, the radius will also become three because radius is mod of G and mod of F, right? Each one of them is actually equal to the radius. Yes or no? So once you know the center and once you know the radius, it is obvious to write down the center radius form X minus three D whole square. Oh, I'm so sorry. X plus three D whole square because, because the center is going to be minus three, minus three, correct? Yes or no? So X plus three D whole square, equal to three square, which on simplification becomes X square plus Y square plus six X plus six Y and plus nine equal to zero. By the way, most of you have given this answer. Well done. You're right. But a small error has happened on your part because you have ignored the second scenario. Right, Nikhil? So G could be negative of F as well. Yes. Most of you have ignored that fact. So let's try to simultaneously solve these two equations as well. Now, since G is negative F, put it in this, you'll get F plus two F is equal to three. That means three F is equal to three. That means F is equal to one. F is equal to one means G is equal to minus one. Okay. And radius is going to be one itself.  You can write down the equation. Many people say, sir, why do you need to, you know, write down the center as well? Okay. Let's use our general form. What is, you know, the general form is there. No, why, why we want to write down the center radius form. So general form, let's take the help of general form. So this will become your, this will become your, this will, oh, C I have to find out. Oh, yeah. I need to use C. I don't know. See, I don't know. So I have to find out the center radius form only. Sorry. I think earlier approach was, we were right on track on the earlier approach. So minus G minus F. So its equation will become something like this, which all simplification gives you x-square plus y-square minus two x plus two of two y equal plus one equal to zero. Okay. So two answers are possible. So most of you went for only one answer. Okay. Let's try to test this on our GeoGibra as well, whether it actually works. But before that, if anything you want to copy down, please do so. Okay. So let's make a scenario here on GeoGibra. So my line was x minus two y equal to three x minus two y equal to three. Okay. And what circles I got? I got, I got these two circles, y-square plus y-square plus six x plus six y plus nine equal to zero. Correct. And you can see here for sure that the center of the circle, center of this, center of equation six, equation six. Yeah. As you can see, it's very well lies on this line. Okay. But we got one more equation. Let's try that out also. x-square plus y-square. If I'm not mistaken, it was minus two x.  Something happens. Sorry, I have to write it again. x-square plus y-square minus two x plus two y plus one equal to zero. Okay. Wonderful. I got another circle touching both the coordinate axes and it's center again on that given line. Okay. Center of A. Yeah. Two equations are possible as what we thought it is assigned any questions, any concerns, anybody, any questions, any concerns. So with this, we move on to another concept that we need to study under circles is the intercepts, intercepts made by a circle on the coordinate axes. So if there is a circle, okay, it may cut the coordinate axes. It may cut the coordinate axes like this. Let me show you one, you know, a rough estimation here. Okay. Let's say this is your x-axis. This is your y-axis. Okay. This length AB, please note this down. Please note this down. AB length is called the x-intercept of the circle or x-intercept cut by the circle. Okay. And this CD is called the y-intercept cut by the circle. Right. So please note intercept in lines was different in lines when we were doing the chapter of straight lines. Intercept was nothing but the directed distance from origin to that point. That was, that was called the intercept, right? Similarly, y-intercept was the directed distance from origin to the point where it cut the y-axis. In case of a circle, the definition of intercept is slightly different. Okay. Here x-intercept is what is the part of the x-axis which is sandwiched by the circle or which is cut by the circle. That is called the x-intercept. Similarly, y-intercept is what is the part of the y-axis which has been cut by the circle, that length. It will always be a positive length. Don't worry. Okay. So how do I find this out? Very simple. Let's find it out. So let me find out the x-intercept first. Okay. Let's say this circle equation is our general form of the equation. Okay. Because normally the question center can give you an equation, any equation of a circle in general form. So one thing that we can say here is that the center becomes minus g minus f, correct? Second thing is the radius becomes, the radius becomes under root of, let me write it in white color just to make it different. Under root of g-square plus f-square minus c. Correct. Now let me ask you a few questions. Let's say this is, this is a center is p. What is this p-m length? Who will tell me? So p-b is your radius, which we already have written on the diagram as well. Okay. What is p-m length? Can anybody tell me that? p-m length? Write it on the chat box. p-m, p-m, p-m, p-m. Modi. Sir, p-m is Modi sir. No Nikhil. p-m, p-m is mod of, yeah, don't forget the mod part because it could be a negative value also. f could be negative also, right? Correct. I see. I copied my joke, Setu. Okay. So how do I find m-b? So we can use the fact that p-m-square plus m-b-square is equal to p-b-square. Pythagoras theorem. Okay. This is your Pythagoras theorem. So p-m is f-square. See, mod minus f-square is f-square only. Okay. No need to do anything else. m-b-square, I don't know. p-b-square is the square of the radius, which is this. So f-square, f-square gets cancelled. So what is m-b then? And since we're talking about m-b as a distance, we will just say the principal root, which is g-square minus c. So what is a-b then? a-b is twice of m-b. Yes or no? Isn't this x-intercept double of m-b? Because we know that a perpendicular from the center bisects the chord. So a-m is equal to m-b. So a-b's will be equal to twice of m-b. So please note this down. x-intercept is given by this formula. Many people remember this formula. If at all they have a good memorizing skills, but I don't endorse memorizing anything. It's better you try to understand this through the proper derivation. If you remember it, you save some time. That's it. And if you practice it, you'll automatically end up remembering it. You don't have to go out of your way to remember anything. Okay. So please note this down. A very interesting question I would like to ask you here. Let us say this is your x-axis. Okay. The circle is, I'm so sorry. The circle is cutting the x-axis, let's say. This is your x-axis. The circle is cutting the x-axis. Correct? What is the value of this intercept? What is the value of this intercept? Positive or negative? You say positive because this is only a positive quantity. Correct? It's a positive value. Right? Good. What about if the circle happens to touch the x-axis? Then what is the intercept here? No, no, no, no. A is not the origin. Yeah. What is the intercept here? Yes. Zero. So intercept here will be zero. Let me write x-intercept. Okay. Here also let me write x-intercept is positive. In fact, I should write it as a real one because you can see it real and positive and positive. But what about a situation like this where the circle doesn't even touch the x-axis? Then what will happen in this case? Oh, let me make it in blue color. I was using blue color only. Yeah. What will happen in this scenario? Right, Nikhil? Now what many people say imaginary, many people say zero for this, but it is actually imaginary. Okay. Please don't make a mistake about this. Okay. So in this case, it becomes imaginary, not zero. Right? So here x-intercept is not a real thing at all because you don't see it intersecting. If you don't see it intersecting, it's imaginary. So many people make a mistake of calling this third case x-intercept as a zero, but no, it is not a zero. It is actually imaginary. So I'm just putting some names to it. So first case, it is real and positive. Second case, it is real and zero. Third case, it is actually imaginary. Well done, Nikhil. Very good. Is this fine? Any questions? Okay. Now I would request you to tell me why intercept. I will not do any further effort over here. Tell me what is going to be similarly calculate the y-intercept and tell me. Write it down on the chat box. Please fill this question not for me. Excellent. How is AB the length of the x-intercept? Setu, that is how it is defined, my dear. That is called that is defined as the x-intercept by definition. Got it. A is here. This is A. This is B. This part, this length is called the x-intercept. That is how it is defined. Very good. Many of you have got this y-intercept also corrects. Please note down, it will come out to be two under root f square minus c. Yes. Same approach. Now I just have to drop a perpendicular from P on to CD and repeat the same activity. Okay. Which I'm sure all of you can do it. Okay. I don't think so. Any assistance is required for that. Okay. So x-intercept and y-intercept please note it down and keep it in case you happen to remember it. You save your time. That's it. But let me tell you, even if you don't know how to find intercept that doesn't stop you from solving any question. Take my assurance. Okay. So it's like, sir, if I don't know x-intercept, y-intercept, does it stop us from solving the question? Not at all. Not at all. Okay. All right. So let's take a quick question on the concept and then we will move forward. Let's take this one. Find the equation of a circle which touches the axis of y. They should have written y-axis, y-axis of y. Yeah, y-axis at a distance of four units from the origin and cuts the intercept of six units from the axis of x. Okay. Good. Let's solve this out.  Give me a response on the chat box. Or if you just want to say a done that is also fine with me. Let's just look into the diagram of the situation. Okay. I'm just doing one of the sample cases. Now you may argue that there could be several cases like this. So I've taken a case where the circle is touching y-axis at four units from the origin. Okay. So this point is let's say four units from the origin. Correct. And this lens that is your x intercept is given to you as six. Let me make it down. I need to form, find the equation of the circles will satisfy this condition. Okay, Nikhil. Okay. From this scenario, can I say, let's say if the center is minus g minus f, let's say I assume my circle is my general form x square plus y square plus 2gx plus 2f5 plus c equal to zero. So first of all, can I say mod of minus f is four. Do you all agree with this or not? Everybody agrees with this or not? Just say yes if you do. And no, if you don't. Yeah. No, people are not saying yes also. Okay. At least one person agrees with me. Nikhil also agrees. Good. So does it mean f could be written as plus minus four. So there are two possibilities of f. Okay. Second thing that I can say is that since it is touching the y axis. This should be zero. Correct. Which means f square should be equal to c, which means c is equal to 16. Correct. And another information given to us is that it's x is equal to c. So the intercept is six units. So to under root g square minus c is six, which means under root of g square minus c is three, which means g square minus c is nine and c is 16. So g square is 25. Which means g is plus minus five. So these equations automatically will reveal all the possibilities for you. So you can see what these questions are. Somebody is looking into all possible diagrams possible. Okay. So what are the circles? It is possible. I could have x square plus y squared plus two gx plus two f y plus c equal to zero. This is one answer. I could have x square plus y squared. Take a minus minus five. equal to zero, then take both of them negative. Then take this positive, this negative. Okay. So there are four answers in all, there are four possibilities. So all of these will be your answers. Okay. So read the question. It is touching at four units from the origin.  Am I right? Yes or no?  Correct?  Right? So four answers are possible and that is what the equation gives me as well. Is this fine? Any questions? Any concerns? Any questions? Any concerns? All right. The next thing that we are going to talk about. Okay. Very good question. So Siddharth, Sethu is asking, sir, had we not remembered the formula, how would I solve this question? Okay. That's even easier to solve actually. See, if you would have not remembered this formula, I would have solved this question like this. Since the circle, since the circle is touching the y-axis at a four distance away from it, I would have definitely chosen one of the cases. Okay. Where I would have claimed that this is some alpha comma four. Okay. Assuming that this is the present scenario right now. Okay. And all of you see here, this is four and this is six. That means this is three. See, this whole thing is six. That means this thing is three. Okay. So that means I would have got my radius as five, which means that this is five, which means that alpha is five. Correct? In other words, I would have got a circle whose center is at five comma four. And radius is five. So write down the equation x minus five whole square, y minus four whole square equal to five square, which is nothing but x square plus y square minus 10x minus 8y plus 16 equal to zero. That would have given you the third case. Similarly, I would have chosen another circle here. Third circle here, four circle here. Four cases I would have got it. Got it. Similarly, similarly, other cases as well. Yes. The four cases will only differ in their science of coefficient of x and y only. Is this fine? Any questions? So with this, we now go on to the concept of intersection of the line y equal to mx plus c with the standard form of a circle, standard form of a circle. What is standard form of a circle? x square plus y square is equal to r square. Isn't it? This was a standard form of a circle. Okay. So now let's say this is my circle, which is your standard form. And let's say y equal to mx plus c is a line. Okay. Now how many ways can a line intersect a circle? It could intersect it like this. In such case, the line is called the secant. You all know that class 10 stuff. Or it could be touching the circle like this. Okay. In such case, it will be called as a tangent. Okay. Or it may even not touch the circle. It can pass without touching it. In such case, we call it as nsnt. Nsnt. Can anybody guess the full form of nsnt? Anybody can guess the full form of nsnt? Let's see how good you are. Right, Nikhil. It's called neither secant nor tangent. Okay. Neither secant nor tangent. It means it is going away from the circle. Okay. Now I would like you to tell me what should be the relationship between m, c and r for these three cases to happen? One by one. Let's talk about one by one. Situation number one. So for situation number one to happen, can I say that the distance of the origin from this line, let's say I call this as d, it must be lesser than the radius. Correct. d must be lesser than the radius for one to happen. In other words, what is d? d is mod c by under root one plus m square. I hope everybody knows how to find distance of a point from a given line. Okay. This should be less than r. Correct. Which means mod c should be less than r under root one plus m square. Correct. Square both sides. So that means c square should be less than r square one plus m square. Okay. So this is a condition that must be satisfied by m, c and r for the line to be secant to this circle. Now either you can remember this formula or you remember the approach. Anyhow, you can solve this question because see a question will come like this. They'll give you a line. Okay. They will say, well, is this line a secant to this circle? So if that line satisfies this criteria, then it will be. For example, if I say y equal to 2x plus one. Okay. And this is the circle x square plus y square is equal to four. Is this line secant to the circle? So what will you do? You just check c square. c is one here. c square. Is it less than r square? r square is four. One plus m square. Is it less than this? Yes. It is satisfied. That means, yes, this is a secant. But in the point, so this condition itself is basically going to tell you whether it is a secant or it is, you know, something else. Okay. Let's take the second scenario. When is this line going to be a tangent to the circle? So you'll say, sir, simple. We don't have to reinvent the wheel. If it is a tangent, d should be equal to one. That means c square should be equal to r square one plus m square. This is a very important condition which we normally call it as condition of tangency. Okay. This is something which you need to use. Remember, because it is useful. What was the distance of a, what are the distance, what is the formula of distance of a point from a line? You tell me, Sethu. And then apply the same. You will get the same result. I have not written some formula coming from, forget, look. Maths is a subject to remember that you forgot. History is not there. History people don't forget. Do you know why? If you understand the chronological sequence, even history, you will not forget. Anyways, for your benefit, I will write it now.  How will you? Yeah. So let's say this is your line a x plus b y plus c equal to zero. Distance formula is given by mod a x 1 b y 1 plus c by under root of a square plus b square. You remember this now? The same thing I use here. Instead of a, b, and c, I wrote it like this. Put zero, zero. So only minus c will be left. Mod of minus c. They write mod of minus c, mod of c means the same thing. By under root of a square b square, a is one, sorry, a is minus m, b is one. Anyways, so please note this down. The third scenario where it is nsnt case, your d would be greater than r. In that case, your c square will become more than r square, one percent. Out of these three, this is more important because a lot of questions will be framed on this concept. When we do, you can say higher version concepts or j concepts of circles. One important thing I would like to bring to your notice is that this condition that you have written down over here or these conditions that you have written down over here, that is only applied when your circle is your standard form, not the general form. If it is a general form, you have to again derive the fresh results depending upon the same concept. So this c square is equal to r square, one percent square is only applicable when your circle is a standard form. What happens is many a times in questions, they will give you the general form and they'll give you y equal to mx per c. So people start using the same formula. Please note you will get a wrong result if you use the same formula because this formula has been derived for this circle, this circle, standard form of a circle. That is why the name of the topic also I have categorically mentioned, standard form of a circle. Are you getting my point? Any questions, any concerns? Let's take a small question based on this concept. For what lambda will this line be a tangent to this circle? Very good, Situ. Anybody else? Okay, Siddharth. Should we discuss it now? Okay, let's recall our condition for tangency. See, first of all, is it a standard form of a circle? What do you say? Yes or no? Yes, it is a standard form. Okay, center is at origin. So now use the condition of tangency. c square is r square 1 plus m square. So here c is lambda, which I don't know. We have to find that out. r square is 5. Okay, remember this is r square, not r.  So can I say it is 5 into 5? That means lambda square is 25. Lambda is plus minus 5. Absolutely right, Situ. Siddharth, why 5 root 5? Any special reason? Oh, yeah, I understood why 5 root 5 because you took this 5 square as your r square, isn't it? This itself is r square. It says self is r square. Okay. Anyway, so we'll stop the concept. This is the end of whatever we wanted to do for circles. At least this will suffice your school level papers.  Meanwhile, tomorrow's session 4 to 7.30 will be taking mathematical reasoning. It is a super easy chapter, very, very easy chapter. Okay. No, no, no, no, no. It just has to be four o'clock. Now there is no online offline. No. High court orders have come. You are at home only now. You ask them to change it now.