Thank you everyone for wishing me the teachers day. Thank you so much. Okay, so we are going to continue with ellipse I knew I had an agenda of taking two hours classes but unfortunately as I told some painting work is going on in my in my house so I will not be able to stretch beyond 930. So this room in fact where I'm taking it it's like half painted. Let's, let's get started. So today I'm going to start with the equation of normals. Okay, so we had already done the candy, we have already done the equation of tangents, we have done the point form, we have done the parametric form, we have done the slope form. And all those forms that I did I would like to reiterate once again that they were only meant for the standard forms. So today I'm going to start with again the standard form of an ellipse our x square by a square y square by b square equal to one ellipse. Okay, and we are going to learn the different forms of the equation of normals done to the ellipse. Thank you. Thank you so much. So let me just pull out my normal diagram I am sure I still have it. No. Yeah, I have it. Yeah. So let's say at a given point, at a given point. x1 y1. I'm drawing a normal to this ellipse. Okay, so we all know normal is nothing but it's a line which is perpendicular to the tangent at the very same point so if I draw a tangent at this point it is going to be perpendicular to the tangent at that point. Okay, so what is the equation of this normal let's try to understand very simple we already have done tangents and normals in our application of derivatives. So this should not be a difficult concept for us so if you differentiate it find d y by dx, you may also use your implicit differentiation formula. So let's say this is our implicit function.  Right. So negative doh f by doh x will be to x by a square. And doh f by doh y will be to y by b square that is nothing but minus b square by a square x by y. If you're finding it at a given point you must substitute then only you'll get a tangent else you will get the gradient function. So please substitute the point in order to get the tangent else that will give you the gradient function. So this gives me minus b square by a square x one by y one. Now having known the point and having known the slope. So slope of the tangent or you can say slope of the normal is negative reciprocal of slope of the tangent. So that is going to become a square y one by b square x one. Okay, so this is the slope of the tangent. So let's find out the equation finally. So minus y one is equal to slope x minus x one. Okay, now we will basically simplify this a bit because we are going to convert it to a particular form. So let's simplify this. Let's multiply. Let's multiply b square on the other side. So you'll have b square y minus b y one. And let's keep. Let's keep. Okay, let's divide this why when down. I think it is b square. So the form that I would like you to note down on your formula list for the equation of the normal is a square x by x one minus b square y by y one equal to a square minus b square. So this is the form that you should be keeping in mind. This is called the point form because here you are trying to find the normal at a given point on the ellipse point form of the normal. Okay, so please note this down a square x by x one minus b square y by y one equal to a square minus b square. Later on, maybe not in today's class maybe in some later classes will be talking about equation of normal at a given point x one y one to a hyperbola. The hyperbola normal equation is very rhyming to this is just that b square will be replaced with a minus b square. Okay, so there will be a plus sign coming here and here. We will talk about it when we start that chapter. So this is the equation of this is the equation of the normal in the point form. Any questions? Could you show the diagram diagram is the customary diagram at a given point x one y one let me call that point as a P. I'm sketching a normal. A square x by x one minus b square y by y one equal to a square minus b square now one interesting mistake which people do here is this is the position of x one and x, but that is a mistake which you can easily catch because in an equation of a line you cannot have the variable in the denominator right so that mistake is easily a wordable if you are slightly attentive. So that mistake I'm just pointing out because many people in the past have done that mistake, switching the position of x x one and switching the position of why why when that mistake happens for many people. So, you can easily avoid that because a line equation cannot be of the nature something by x minus something by y equal to constant that will be a wrong equation of a line. Okay. Having taken the point form that is now move on to the parametric form of the equation of a line. So let's move on to the parametric form parametric form of the equation of a line. See in parametric form is just that. Let me just pull the diagram back again in parametric form we are going to learn what is the equation of the very same normal, but just that now the point is express as a parametric point you can say a cos theta B sin theta. Okay, so this point P has an eccentric angle of theta so what is the equation of the normal over here. So nothing different over here all you need to do is just take the old equation replace your x one with a cos theta replace your y one with also why one with B sin theta. So when I do that, I'll end up getting a square x by a cos theta minus B square y by B sin theta equal to a square minus B square. On slide simplification, your result will look like this a x seek theta minus B y cosec theta equal to a square minus B square. Now a very interesting question is framed on this particular fact. So everybody please note this down. So this is the equation in parametric form. All these equations which I'm giving you whether it is a point form or whether it is the parametric form they are applicable to a standard case of an ellipse please do not apply it to any non standard cases. Okay, of course you can apply it but only by accounting for the shifting and everything that you basically take care of when you are trying to solve a non standard case. So there you can apply but only along with the shifting concepts that you have already learned. Now, before going on to the slow form there's a small question I would like you to answer. If y is equal to mx plus C is a normal to, let's say our standard form of ellipse x square by a square plus y square is equal to be square. Then show that C square is going to be m square a square minus B square whole square upon a square B square m square. Okay, by the way, this result that you see over here, this result is also called condition of normality. Just like there's a condition for tangency, that is C square is equal to a square m square plus B square for a line y equal to mx plus C to be a tangent to a standard form of an ellipse. In the same way y equal to mx plus C to be a normal to a standard form of an ellipse. The condition for normality is C square is equal to m square times a square minus B square whole square upon a square plus B square m square. Everybody please try to prove this. It's very easy. And do let me know with a done that you are done with it. See, again, the process is very well defined. You have already got this equation know a seek theta minus by coseak theta is equal to a square minus B square, right. And we are basically trying to compare this equation with y equal to mx plus C. So let me write this equation like this mx minus y is equal to minus C. Okay, now we need to compare these two equations because I asked for the question the seek these two equations are the same because they represent the normal right. Yes. So, so what I have to do here I have to achieve this condition of normality by eliminating theta here. Okay, so let us do that very simple process. So, let's take the ratio of the coefficients so m divided by a seek theta minus one divided by minus by coseak theta rather I will write it just as one divided by B coseak theta and minus C divided by a square minus B squared. Okay, so from here you end up getting cos theta times m by a sin theta by B is equal to negative C by a square minus B squared. So from here cos theta is going to be negative AC by m a square minus B squared and sin theta is negative BC by a square minus B squared. Now we know our famous Pythagorean identity that cos square theta plus sine square theta is going to be one. So let us use that. Okay, so a square C square m square a square minus B square whole square, then B square C square a square minus B square whole square is equal to one. Okay, so first of all I would send the a square minus B square whole square to the other side. So you'll have something like this left. So here also you can just do a quick simplification nothing very challenging here. And there you go. A square is equal to m square a square minus a square minus B square upon a square plus B square m square and there was a square here also let's not forget that this is the condition for normality. This is the condition for our condition of normality. Is it fine any questions. Wasn't it easy. Yes, any questions. Yeah, it should have been done. Yeah, it was easy actually. Okay, let's have few questions one or two, and then we'll move on to a very interesting concept called the co normal points. Anything that you would know would like to know down from here please do so I'll be shifting my screen. So basically simple with the parametric form I ended up comparing y equal to mx plus C. Okay, so from here one more thing comes into picture that if I have been provided with a slope. If I've been provided with a slope, if M of the slope of the normal is given to you, then the equation of the normal can also be written as for a standard form of an ellipse, why is equal to mx plus minus m a square B In fact, it should come out as a mod but I've taken a plus minus, not an issue. Okay, so this can be called as the slow form of the equation of the normal. So we had a point form we had a parametric form now you have a slow form of the equation of a normal. Okay, let's take few questions. Simple one maybe I've already done a similar question a little while ago. But you can still try prove that the straight line Lx plus my plus n equal to zero is a normal to this ellipse. If this condition is met, just let me know with a done that you are done with it. See, they don't need to actually memorize the condition of normality you can proceed solving this question in the very same way as how I achieve that condition of normality. So please do not try to remember things which are unnecessary. So, solve it from the basics. See, you can start with the equation that we have discussed for the parametric form. Okay, so this was the equation the parametric form a x theta minus b y cosec theta is equal to a square minus b square now you're trying to claim that this line. This line is exactly the same. Excellent. Okay, so same way, compare the coefficients. So exit theta by L minus be cosec theta by m is equal to a square minus b square by minus m. Isn't it. So let's write, seek as reciprocal of course, let's write cosec as reciprocal of sign and use your famous Pythagorean identity to eliminate theta here. And sine theta would be very good and it'll cost theta would be minus na by L na by L by a square minus b square and sine theta will be nb by m a square minus b square. Do let me know if I have missed out on any expression here I don't think so I have taken care of everything yeah. So the fact that cos square theta plus sine square theta is going to be a one. So this is going to be n square a square by L square a square minus b square plus n square b square by m square a square minus b square whole square. That's going to be a one. Okay, so just simply take, simply take a square minus b square whole square by n square to the right hand side. And this is what we wanted to prove. This is fine. Any questions. Okay, with this we now move on to the concept of co normal points. So co normal points concept we have already seen in our parabola chapter that means from a point we can actually draw three normals to the parabola. We had also seen the condition where the parabola will be having a real form of the equation of the normal or how many real cases can be formed depending upon where is the point actually line. Okay. In case of an ellipse also will be learning a similar set of condition. Let me pull out the diagram. So from a point, let's say h comma k, from a point h comma k, you can actually draw four normals to an ellipse. So let's say I take the point to be h comma k here. Okay, so from this point I can actually draw. I can actually draw four normals to this ellipse. Okay, I'm just doing some orbit normals here. So maybe something like this, something like this. Now, first of all, how four normals. Okay, why four normals, why not lesser than that and why not more than that. Okay, let's try to analyze that first. Okay. So here a b c d, they will be called as the co normal points, co normal points. The feet of the normal, the feet of the normal drawn from this point h comma k to the ellipse. They will all form a co normal point. Okay, and of course they can be various such set of four co normal points right so this is just one example which I have shown you. Okay. So we'll first understand why four co normal can be drawn and why not more or less. Okay, let's try to look into the equation of the normal to understand that. The equation of the normal used to be a x seek theta minus by co seek theta equal to a square minus b square a square minus b square this was our equation of a normal, isn't it. Of course, the all the normals pass through h comma k so h comma k must satisfy, must satisfy the above equation. Let me call this as one. Okay, so if it does I will have a hc theta minus bk co seek theta equal to a square minus b square. Now, listen to this simple analysis of mine. This theta that you have in the question. This theta is a generic representation of the eccentric angle of the foot of the perpendicular right. So this theta this theta is a general representation general representation, the eccentricity. I should say eccentric angle I should say eccentric angle of a b cd. Correct. That means, let's say this is in reality alpha this is in reality beta this is in reality gamma this is in reality delta, but I don't know whether they will be four or not. So as of now, I'm not, you know, commenting on that, but I'm what I'm trying to say is that this theta is a general representation of the eccentric angle of the foot of the perpendicular. So the number of thetas will actually tell me the number of thetas will actually tell me how many foot of the perpendicular are getting formed. Am I right. So, the number of thetas that I get, they will indirectly tell me how many a b cd fgh I mean I'm just thinking from thinking from a generic point of view. So how many feet feet of the perpendicular is going to get formed that will be dictated to me by how many thetas I will get. Am I right. Yes or no. So what I'm going to do is I'm going to use a bit of complex number idea over here. How see. Let us say this theta I associate with a complex number. So let Z be. Okay. Can I do that. Can I associate. Can I associate the eccentric angles with a complex number. I can definitely do that. So cost theta minus I sign theta is one by Z. So if you add them you get to cost theta as Z plus one by Z. That means cost of theta is half of Z plus one by Z. Correct. Similarly, if you subtract them, you'll get to I sign theta as Z minus one by Z. That is to say that sin theta is one by two I Z minus one by Z. Now what I'm going to do is I'm going to substitute these two in our equation of the normal. Okay, so I'm going to substitute cost theta and sin theta in terms of Z. In the equation of the normal and see what is the degree of Z which I get. The degree of Z will be indicative of how many normal or how many feet of the normal points indirectly how many eccentric angles are given by this theta. So let's try to put that. So when I do that, I will end up getting something like this, a H by half, I can strike it like this. Instead of writing half that present I just combine it as that square plus one by two Z minus BK. And this will be Z square minus one by two I Z is equal to a square minus B squared. Is it fine. Let's do a bit of simplification over here. I think this will be to a H Z by Z square plus one and this will be minus two I BK Z by Z square minus one is equal to a square minus B squared. Okay, let's do a further simplification. To a H Z Z square minus one minus two I BK Z Z square plus one and LCM of this will be Z to the power of four minus one that I'm sending to the other side. Okay, let's simplify it. So here if I make a equation out of Z I will get a square minus B square Z square. Okay, I will end up getting minus two. Just correct me if I'm wrong I'm just collecting the Z cubed terms. Okay, so you get minus two a H minus I BK Z cubed. So I'm just collecting Z squared Z squared term is this anywhere no Z squared term okay so zero Z squared. Good. And what about Z term so Z term will be obtained from here and hit so it'll be to a H plus I BK Z and constant term will be minus a square minus B squared equal to zero. I've just simplified this. Nothing else I've just simplified this you can also verify that the simplification is correct. Anything that I'm missing do let me know minus a H, which one you're talking about. Proficient of which term here X Z Z term. No Z term will come on the right side it will come to a HZ and minus IBK Z. I'm bringing these two terms to the right side. Yeah, no issues, no issues. Okay. Now what does this fourth degree expression or fourth degree polynomial in Z tell you that there are four possible z so this is a fourth degree expression. Fourth degree expression implies that there will be four roots to it. Correct. And if there are four roots to it for each of the roots, there will be one. You can say argument, and that argument is nothing but your eccentric angles for those corresponding points that's alpha beta gamma delta. Okay, so let's say these are your arguments of arguments of your roots, Z 1 Z 2 Z 3 Z 4. So every argument basically signifies that there is a point on the ellipse where these normals are going to meet. In fact, indirectly speaking, that there are four points from where if you draw normals they will be concurrent at H comma K. So this signifies that there will be four co-normal points. There will be four co-normal points. Is this fine is this analysis understandable. There are a bit of complex numbers with coordinate geometry here. So this four point shows that there are four eccentric angles. See, when it comes to eccentric angles, basically I tried to compare it with argument. There are other ways also to do it.  That's another way to do it. It's just like saying that if I have to have four angles. Okay. That means if I make those four angles alpha beta gamma delta to be your argument of four complex numbers and if I somehow show that it will be a fourth degree expression in Z, then I'm achieving the same thing am I not. That's the only purpose of doing it. So I was trying to relate eccentric eccentric angle to the argument. But let me tell you, this is not the only way to do it. Many people, as I told you, they write tan in terms, sorry, they write cos and sine in terms of half angles of tan, and they get a fourth degree equation. Now why half angles of tan because see this is very important to understand. If you get, if you just write it in terms of sine or cos or just a tan, you will end up getting two such angles for which tan will be the same. There by giving you a incorrect image that there were two distinct points not one. That is why when you convert it to half angles in tan, when you take it as a half angles, you have to lie between zero to pi and they will automatically different sign and they will tell you the reality. Right. Are you getting my point that is why many people converted to half angles of tan and do the process. But I find this one more convenient because related to this there are two properties which we're going to see in some time and we're going to prove that very easily when we use the complex number notation. Okay, so let us take a few properties based on the co normal points. Meanwhile, please note this down. I mean, don't have to remember this equation but just have just be aware of the fact that there are four co normal points for case of an ellipse. Unlike in case of a parabola they were, they were only three. So let's take, let's take a simple, you can say property based question. So it's a question also and it's a property also. Let's take this one. In general, four normals can be drawn to an ellipse from any point. And if alpha, beta, gamma, delta, the eccentric angles of these four co normal points, then please prove that alpha plus beta plus gamma plus delta will always be an odd multiple of pipe. This is a property of co normal points properties of co normal points, prove that the feet of the perpendicular, the feet of the perpendicular of these normals. The eccentric angles will add up to give you an odd multiple of pipe always. If you want, I can rewrite the equation once again that I had written in the previous slide. I think it was minus two h minus ibk right. Ibk, z cube, there was no z square term. So zero z square and two h plus ibk. Tell me how will you do this? The answer to that this question or this property is hidden in this equation itself. If you want to add the arguments of a complex number, what do you do normally? Tell me, go back to your class 11 days and tell me if arguments are to be added of complex number, what operation should happen between them? You need to multiply them. So if I ask you what are the product of the roots of this particular bi quadratic equation in z, what will you say? What are the product of a root minus b by c by d by minus e by a. So minus e by a. So let me just write it like this. Okay, so if all the roots product are required minus b plus minus plus, yeah, so it will be plus e by a. A here is also a square minus b square. Yes or no? So I'm basically using my beta isolation. So in beta isolation, the product of the roots of a bi quadratic equation is the constant term divided by the coefficient of z to the power four. Right. Some will be minus b by some of product two at a time will be C by some of product three at a time will be minus d by product of all four will be e by so that's minus one. Right. Now let us say each of these complex numbers z one is made up of is made up of such arguments which are the eccentric angles of those points. So the product will be what and if you want you can write it like this also not an issue Oilers notation. Okay, Oilers notation of complex numbers. So this is nothing but e to the power i alpha plus beta plus gamma plus delta that is equal to minus one. Okay.  Minus one is minus one is minus one plus I zero. So you are comparing two complex numbers here so which clearly implies that this is going to be this is going to be minus one and this is going to be zero. Now what is the only possibility which can give this. This can only happen this to can only simultaneously happen when your alpha plus beta plus gamma plus delta is an odd multiple of five. Isn't it. Isn't it for all multiple of pie cost will be a negative one and sign as we know that for any multiple of pie sign gives you zero. That's it. That's proved. Okay. What happens any concerns with the approach. See, normally eccentric angles are between zero to two pie. No. So alpha plus beta plus gamma can go from zero to eight by maximum. Yeah. All right. So next property, again, I mean it's based on the very same fact that you have derived. This property you can easily prove it by using the previous result itself. If alpha beta gamma are the eccentric angles of three points. So please note that they are just referring to three of the points. Okay. So alpha beta gamma are the eccentric angles of three points on the ellipse the normal set which are concurrent then show that or prove that this is going to be zero. The hint is, hint is use this and let me know once you're done. Done. Okay. I mean, it's not that that simple also. But yes, let's discuss it out. Front of says he is done. Okay, see everybody. Let us not forget that the coefficient of that square is actually zero. This is a very important part. Okay. Coefficient of that square is zero. What does it mean. It means that the sum of the product to at a time is equal to zero. Isn't it. That means Z1 Z2 Z2 Z3 Z3 Z4, let's say Z1 Z2 Z3 are the roots of that by quadratic equation in Z. Z4 Z1. I think Z1 Z3 I missed out and Z2 Z4. Yeah. That means this should be equal to zero. So what does it tell you that if you start writing your Z1 as e to the power i alpha Z2 as e to the power i beta Z3 as e to the power i gamma and Z4 as e to the power i delta. So can I say that e to the power i alpha plus beta e to the power i beta plus gamma e to the power i gamma plus alpha e to the power i delta. E to the power i delta plus alpha e to the power i alpha plus gamma and e to the power i beta plus delta this should be equal to zero. Because the coefficient of Z2 in the by quadratic was zero. Okay, so I'm just writing it down so that when you're referring to your notes, let's say little later on. Okay. So, what does it mean indirectly is that cause of alpha plus beta. I mean just letting, letting it down like this this will be zero and summation of sign alpha plus beta this will also be zero. In fact, from this particular concept these two properties also evolve. So please note down as a property itself that if you take cause of any two of the angles are some from those eccentric angles, which are the feet of the co normal points, the cause summation will always give you a zero. Similarly, sign summation will also give you a zero. Okay, now what I'm going to do I'm going to use this result to achieve my purpose. I'll see everybody sign alpha plus beta. Okay, let it be as it is sign beta plus gamma let it be as it is then sign gamma plus delta. Okay, gamma plus delta sign delta plus alpha sign alpha plus gamma and sign beta plus delta is equal to zero. I will not disturb this term let it be as it is, because it is there in my expression. I will not disturb this term I will let it be as it is because it's there in my expression. I will not disturb this term also. Where is, where is gamma plus alpha. Okay, there's no gamma plus alpha. Not possible. Yeah, I will not disturb this term also. Okay. So what I'm going to do is this term, this term. So there are six terms on all together right. Correct. So this term is also taken care of. Yeah. Now how do I take care of these facts. What will happen to the other three terms. Sign of no down sign of gamma plus delta is same as sign of an odd multiple of pi minus alpha minus beta why because alpha plus beta plus gamma plus delta was an odd multiple of. Pi, correct. So instead of writing gamma plus delta, I wrote it as 2n plus one pi minus alpha minus beta. Okay, and when you take sign of both of them, you will realize that this will automatically give you sign alpha plus beta. Right. Similarly, so this becomes this term becomes this term. These two are same terms. Correct. Similarly, you would realize that a delta plus alpha will be same as beta plus gamma. So this and this will be same terms. Correct. And this and this will be same terms. In short, you have written the three ticked terms twice here. Are you getting my point, which means you have written two times sign alpha plus beta sign beta plus gamma. And sign gamma plus delta two times, and you're claiming it to be zero, which means, which means drop the, you know, the two factor. So which means sign alpha plus beta. Sign, let me just drag it over here a bit. I think I have space. This is equal to zero. Hence proved. Okay, so from this tool, other properties also came up. That is cause alpha plus beta summation for all possible values of alpha and beta is zero sign alpha plus beta is also zero summation of that. And moreover, because of the fact that alpha plus beta plus gamma plus delta is an odd multiple of pi we also came into the fact that even if you take any three of the eccentric angles, let's say alpha beta and gamma in the case. And this term will also be giving you the zero expression. Is it fine. Anything that you would like me to repeat over here do let me know. So first I use the fact that since there is no z square coefficient or that square coefficient is zero. The sum of the product to what the time is zero, which led to these two results from the second result I wrote this expression in yellow. I did not disturb the terms which were already present in my given expression that is these three terms. I've ticked them on the top. Okay. And the rest three terms, they actually convert to the ticked terms only. So sine gamma plus delta is as good as sine alpha plus beta. Right, because of this hint which I gave you. Sine delta plus alpha is same as sine beta plus gamma. And sine beta plus delta is same as sine alpha plus gamma. And that's why this entire sum becomes twice of the given expression. And which is anyways going to be zero because this is zero. So the property holds true. Is it fine. Any questions. These feet of the perpendicular that means these go normal points. That's the third property. The feet of the co normal points. They lie on a curve. Okay, whose equation is, whose equation is a square minus B square x y. Let me write it down like this the feet of the co normal points. Concurrent at h comma can lie on a curve. A square minus B square x y plus B square kx minus a square h y equal to zero. Please prove this. Please prove this that the feet of the normal they lie on a curve which is given by this expression. Everybody please prove this. See the proof is very simple. I mean, just, it's just a, you know, observation that you have to do. So let me just pull out the diagram here. Okay, so let us say, let us say, this is my h comma k point. This is my feet of the perpendicular. I'm not getting. No worries. No worries. See, first of all, what is the equation of the normal that we know of a square x by x one. Okay. Minus B square y by y one is equal to a square minus B square. Okay. Now, as per this particular condition x one y one is the point. This is the, this is representative of the coordinates of the point. And this normal should pass through h comma k, right? So if you substitute your, if you substitute your h with x or x with h, y with a k, then this is the equation that I would be getting, isn't it? In short, what do you have written over here is a square a square h y one minus B square k x one is equal to a square minus B square x one y one. Okay. In short, you have written something like this a square minus B square x one y one plus B square k x one minus a square h y one equal to zero. Right. But can I say the same will be true. This will be true even if your point is x two y two x three y three and x one y four. Right now you started with this point as your initiating point you started with this equation. Then you made h comma k, you know, satisfy the equation. But the same will be true even if you have taken B as your starting point that means you would have started with your a square x by x two minus B square y by y two is equal to a square minus B square. And still h comma k will satisfy that equation of the normal, isn't it? Do you agree with me on this or not? This is very important. This will be true even for these three points other than x one y one. So x one y one is definitely satisfying it even x two y two will satisfy it even x three y three will satisfy it and even x four y four will satisfy it. Am I right? Are you convinced with that? Correct. So now just have a look at it even if you replace your x one y one with x two y two. Okay. And further replace it with x three y three, further replace with x four y four. Correct. Now what is the structure of this equation? This basically equation is trying to say that I'm so sorry. Yeah, this structure that you see over here, these four expressions that you see here, what do they seem to suggest? They seem to suggest that if you replace your x i y i's with your x y, you will end up getting a curve which is going to be satisfied by all the four points and that curve is this. So this equation will be satisfied by, let me write it here, this equation will be satisfied by all these points which are your feet of the normal. Correct. And this is the equation that we are basically asked to prove over here. Correct. So the feet of the co-normal points or you can see the coordinates of the co-normal points or the feet of the concurrent normals to an ellipse, they will pass through this curve. Okay. So this itself can come as a question to you in your comparative exams. By the way, this is given a special name. This is called the Applonian rectangular hyperbola. Applonian rectangular hyperbola. We use this word Applonian circle in case of a complex number last year. Correct. So a complex number which basically satisfies this locus condition, where k is not equal to one, then z will trace a Applonian circle. Right, remember, I don't know how many of you remember that last year we had done complex number. So basically the name of Applonian has come from a very, very ancient mathematician, Applonius. So just to honor his name, this was given, maybe he had done some work on this field, that's fine. So I'm not very sure about the history behind it, but this is what we call as the Applonian rectangular hyperbola. So the feet of the co-normal points, feet of the concurrent normals or the co-normal points satisfy this curve which is called Applonian rectangular hyperbola. Exactly, exactly. So this rectangular hyperbola will actually satisfy this four co-normal points will actually satisfy both ellipse equation and this rectangular hyperbola. Why don't we do one thing? We can try to figure it out, but of course we have to ensure I've taken a point h comma k. Okay, maybe we'll try it out, you know, offline. Some of you please try this out and see how does the diagram look like. Okay, so I'll just write it down, try it out on GOG. And share on the group if at all you are going to get these curves and so that everybody sees them. So you have to choose your point h comma k very carefully so that you can draw that four co-normal points. The idea to make co-normal points is choose your eccentric angles first such that there are some of the angles of the eccentric angle is coming out to be an odd multiple of pi. From there you sketch four normals, they will be concurrent at h comma k rather than doing it the other way around. Okay, next concept that we're going to talk about is very similar to what we have already done in circle. So in circles we had done concepts of pair of tangents. Okay, drawn from x1, y1. So if you remember, the concept was T square is equal to SS1. So this used to give you the pair of tangents drawn from an external point x1, y1 onto the ellipse. So let me just pull in our diagram quickly over here. So let's say from this external point x1, y1, you're drawing two tangents. So this pair of tangent equation is given by T square is equal to SS1. I hope you're all aware of what is your T expression. So if I talk about a standard case, okay, if I talk about a standard case, this is what we call as S. This is what we call as a S1 and this is what we call as a T. I hope you all are well aware now of, well aware of now the basic expression for SS1 and T. Okay, maybe a simple question we will take up on this. Let's say we have this ellipse. Okay. The question is, find the equation of pair of tangents, pair of tangents drawn from, let's say 0.4,3. Let's say 4,5 to be more, I'll say 5,6 to be more precise because I want to be outside the ellipse. Just to get a hands on on this formula, that's it. I don't have any other intention. Just to get a hands on on this formula, T square is equal to SS1. Let's just take this problem. No need to simplify it. Just write down the expression and leave it. I mean, you can always simplify it if the need be, but just write down the expression and just let me know by saying and done on the chat box. Done, Anusha is done. Okay, simple. So what is your T expression here? T expression will be X, X1 plus Y, Y1, okay, by a square minus 1. So whole square of this is equal to S and S1. S1 will be just replace your X with a 5 in the equation of the ellipse and Y with a 6 in the equation of an ellipse. I'm not simplifying it. Okay. So whatever equation you get from here, this will be a second degree equation which will represent pair of straight lines or pair of straight lines. Or pair of tangents. Is it fine? Any questions? Now, many times students ask me, sir, if I want to get the two tangents separately, then what will I do? One way is to basically factorize this second degree equation, which I will be telling you when I do pair of straight lines concept with you. Other would be you start the event. Once you know this point, you can start the fact that let the tangent be this. Okay. So once you start with this as a tangent equation, use the condition of tangency. Okay. Then use the condition of tangency. C square is equal to a square m square plus b square. You'll end up getting a quadratic in m. Okay. Solve the quadratic to get two values of m. Put it over here. You get the two tangents separately. I'll repeat whatever I've said. Okay. So please pay attention. If let's say the requirement be you want the two equation of the tangent separately. Okay. So what do we, what do you do for that? So you say, you assume that the tangent equation is y minus y one is equal to slope x minus x one. Remember y one and x one values will be given to you. So why one and x one values will be given to you. So this is the condition. This is the tangent. And this is the equation of an ellipse.  This will give you a quadratic in m. Okay. So solve the quadratic. Solve the quadratic in m. You'll get two values. Okay. Those two values you put it over here. Those will be giving you the two tangents approaches here. How to find two tangents separately. Other concepts like equation of chord of contact. Let me just take up one more concept. Since we have discussed about equation of pair of tangents, director circle, director circle. So what is the locus definition of director circle? Director circle basically is the locus of points from where perpendicular tangents. Let me write it down from there. Two tangents can be drawn to the curve drawn to the curve, which are perpendicular to each other. Okay. So the very same diagram. Now I'm drawing it again. Oh, sorry. So if you have, if you have a point from where you're drawing two tangents, which are perpendicular to each other. Okay. Let's say this is my point P. Okay. Then the locus of all such points, the locus of all such points. Okay. That will be the director circle. And of course in this case also it will trace a circle only. So let me just draw a circular path around it. Now I would request you all to give me the equation of this director circle. Okay. So what is the equation of this director circle? So let's say this is a locus question which was given to us in let's say J main exam. Okay. So this is a standard case of an ellipse x square by a square y square by b square equal to one. And I want to find the locus of all such points from where if you draw tangents from where if you draw tangents to the ellipse. Okay. Any point you take it draw tangents to the ellipse that those tangents will be at 90 degrees to each other. So what is the equation of this director circuit? Please solve this and give me a response on the chat box. This will also help me to test your understanding of how to solve locus questions. Okay. Should we discuss it? All right. So let us say this slope is M. Okay. And it is passing through H comma K. So you can write down the equation of you can write down the equation of the tangent to be this. Okay. Or in short, y is equal to mx minus or you can say ms plus K minus mh. Okay.  This is a tangent to tangent to tangent to our ellipse. Okay. Now, one important thing I would like to highlight over here. Many times people argue with me sir, why do you take H comma K in locus questions and then you substitute it with X and Y. See here, there's an already X and Y sitting in our tangent equation. And if you slide your H in K again with an X and Y that will create a lot of course. Right. So I think last week only I was teaching locus to your juniors. So in one of the batches, one student was quite irritated. He said that sir, why do you take H comma K and then replace it back with X and Y. In the last step, why don't you take X and Y in the beginning itself and save your time.  But later on when you do conics, there will be a lot of instances where X and Y is also used and H and K is also used. And then you will get confused. Okay. So it's very important that we follow the norms that is, you know, discuss with us. Anyways, if this is a tangent then condition of tangency must be fulfilled. So what is the C here? This is the C right. A square M square and V square that is not going to change. Now this is a quadratic in M. This is what I was basically talking about. If you write this as a quadratic in M, you will have K square minus 2KMH plus M square X square equal to a square M square plus B square. So this will be M square A square minus X square. Okay. And you'll have plus 2KMH plus B square minus K square equal to zero. Right. Now in this quadratic, you have two roots M1 and M2. Right. Let us say, and you want the, you want the two tangents to be perpendicular. So let's say if this is M1 and this is M2, and you want them to be perpendicular, clearly M1 M2 should be negative one. Isn't it? Very good. That's the right answer. So here M1 M2 is going to be minus one. M1 M2 is nothing but the product of the roots of a quadratic, which is C by A. We all know by beta isolation, this is equal to C by A. So this should be minus one. That is to say that B square minus K square is X square minus A square in short, X square plus K square is equal to A square plus B square. Now here you generalize your edge with X and K with a Y. That gives you the equation of the director circle to be this. Please note that this radius is even more than the semi major axis lens. So it is even more than that. It is under root of A square plus B square that is the radius. So radius of director circle is radius of director circle is and the center of the circle is same as that of the conic. Okay. So this is in general applicable to any type of an ellipse. Center will be same as that of the center of the ellipse, wherever it is, whether it is shifted, whether it is oblique and radius will always be under root of A square plus B square. So this is true in general. Is this fine. So before we move on, we will like, I will like to discuss with you. Equation of a chord bisected at a given point X one Y one. This concept is also same as what we learned for a circle and parabola, which is T equal to S one. Okay. So if you have been given a conic, if you've been given a conic X square by a square Y square by B square equal to one. And there is a chord whose midpoint is known to you. Let's say this midpoint is known to you X one Y one. Then the equation of this chord is given by equal to S one. And this, it is very easy for us to cancel out minus one and minus one, but just for remembering purpose. So that this is easy to remember, isn't it. So you can just carry that extra burden, even though it will get canceled from both the sides while you are trying to solve this problem. But this helps you to remember it T equal to S one. So I will not be going into the derivation, etc, because the result is more important for us. And we'll just take a question based on this. Again a locus question. The question says tangents and right angles are drawn to the syllabus. Show that the locus of the midpoints of the chord of contact is this curve. Beautiful question. Next when we meet we are going to talk about equation of the diameter and something related to conjugate diameter. I think that's the only concept which is left off conjugate diameter is also important. A lot of questions in J advance have come on that. So we'll talk about that. And then we can close this chapter and move on to hyperbola. Okay, so basically you're going to tangents from it, which are at right angles. Okay. This is your chord of contact. Okay. We have to find the locus of the midpoints. So let's say this is point P. Let's say this is AB. So we have to find the locus of this guy.  So if you keep changing this P position on the director circle, obviously the chord of contact will also keep moving and the midpoint will also keep moving. And we have to find the locus of the midpoint. Okay. Let's solve this question in this way. Let's say this point is X1 Y1. Okay. Now, what is the chord of contact when you draw two tangents from X1 Y1? The chord of contact is T equal to zero. We have already seen it. So that means your equation of AB, equation of AB is going to be XX1 by A square, YY1 by B square equal to one. Now, at the same time, be very careful. X1 Y1 basically lies on director circle. We will use this little later on. As of now, we'll keep it in terms of X1 Y1. Same, if you find the equation of the chord whose midpoint is H comma K, the equation is T equal to S1. Even this will give you the equation of AB only, which is T equal to S1. And that is equal to HX by A square. I'm just writing T expression, KY by B square minus one that will eventually get cancelled. So that's not waste time writing that. Okay. Now here, these two equations represent the same things. This is also equation of AB. This is also equation of AB. Okay. So let's compare one and two, because both represent equation of AB only. So if you compare the coefficients, let's compare the coefficient of X by A square on both sides. So X1 by H will be equal to Y1 by K. Please note that other terms will get cancelled off. So there's no point writing it down. One by H square by A square plus K square by B square. So this is a vital step for us because after this our problem is almost solved. How? Your X1 is H by this and Y1 is K by this. And this must satisfy the director circle equation, which is just now seen as X1 square plus Y1 square is equal to A square plus B square. So we're just putting these values in place of X1 and in place of Y1 over here that will lead to H square by H square by A square K square by B square whole square. Similarly, K square by H square by A square K square by B square whole square equal to A square plus B square. Okay. So this is really close to the expression. Yeah, so almost done, almost done. So S square plus K square, you can just, you know, take one by this common out. Maybe after the class you can try sketching it also what kind of a curve comes out from it. Okay, and then you can just, you can just make it on GOG brand check. Okay, now generalize it. After this you can generalize it. Okay, X square by A square, Y square by B square whole square as X square plus Y square by A square plus B square. This is the desired locus. Okay. Now this chapter is still not completed as I told you a concept of diameter conjugate diameter will also talk about bit of consicclic points also. In the next class I will be definitely finishing it off and maybe I will also start with hyperbola whenever we meet next. Okay. Thank you class.


Hello everybody. Thank you so much for coming to the panel. My name is Annette. I'm a sophomore here at Cal. I'm studying film and art practice. In my free time, I like to volunteer with the UC Rally Committee and we get to host a ton of really fun spirit events for students on campus and also introduce all the lovely panelists that you're about to hear from. Before we get started on the panel, I want to make sure to thank all of our generations of donors for all of their contributions to the university and the opportunities that they give all of us students here. And then now to introduce the panelists. We have Katherine Carson who's the chair and professor of history at UC Berkeley. She is a specialist in the history of 20th century physics including its cultural, social, and political contexts. She co-edited a volume of papers about Oppenheimer re-appraising Oppenheimer's centennial studies and reflections that was presented during a 2004 conference celebrating the 100th anniversary of his birth. Next we have Yasunori Nomura, a physics professor and director of the Berkeley Center for Theoretical Physics. Professor Nomura works on theoretical physics, in particular theoretical particle physics, cosmology, quantum gravity, and quantum information. His current research focus includes black hole information problem, holographic descriptions of space time, and multiverse cosmology. And then finally we have Carl van Biber, a professor of nuclear engineering and the executive associate dean of the College of Engineering. Professor van Biber studies basic and applied nuclear science, particle astrophysics, and accelerator science and technology. He spent 25 years conducting nuclear energy research at the Lawrence Livermore National Laboratory. So thank you, everybody, for coming. Good afternoon. This is clearly an only at Berkeley panel. And it's wonderful to see you all here today. Can I first get a show of hands? Who has seen the Oppenheimer movie? Okay, that's about 80%. And then there's the 20% remnant who haven't seen it yet, but hopefully that will be an inspiration to you coming out of this afternoon. It's wonderful also to have this panel today. Robert Oppenheimer's years at UC Berkeley were absolutely accrucible for him. They were critical moments in his life. And this campus and its larger setting, its environs were at the center of his life for a crucial decade and more. So to start off, let me set the scene a bit for you. So you probably know that Oppenheimer was the brilliant child of a New York City highly cultivated family in the turn of the century era. He skipped grades in elementary school. He spent his high school years looking at literature and languages and arts and science. And as a college student at Harvard, he studied chemistry and then physics. And he followed that muse on to graduate school in Europe for his PhD. That was what one did back then. So he landed in the discipline of physics at a time of revolutionary change when the scientific revolution around quantum mechanics was really coming into its own. This was what he made his own specialty, applying quantum mechanics to molecules, to the nucleus, to stars. And he also took it further, trying to figure out how to generalize quantum mechanics to something called quantum field theory. That was the frontier of his discipline in the years that he was at Berkeley. So he learned not just to apply quantum mechanics, not just calculate, but also think. Think quantum mechanically. Think in this deeply abstract and counterintuitive way that modern physics has at its core. That became one of his signature strengths that he carried with him after he moved beyond simply the work of a theorist. Now you're probably used to thinking of theoretical physics as a central discipline in modern intellectual life. And as a historian, I'm here to say it wasn't always that way. In the 1920s, when Oppenheimer chose his life's work, theoretical physicists were really curious creatures, very few in numbers, and largely removed from the world. That would change fast over the course of basically those two decades, between 1925 with quantum mechanics and 1945, the end of the Second World War. So Oppenheimer's life unfolded in the midst of that transformation, and in important ways he helped drive it at Berkeley. Because when the Berkeley Physics Department made the decision in 1929 to hire him, it hired the best trained, most intellectually adventurous theoretical physicists in the United States. And Oppenheimer set out here at Berkeley to build a school of theoretical physics to cultivate the next generation here in California. And he did. He built the best school of theoretical physics hands down in the United States. His success marked the department here at Berkeley, it marked the campus, and permanently it stayed even after he departed for Los Alamos, and then after the war to the East Coast corridors of power. Oppenheimer is at our center today, but he and his colleague Ernest Lawrence were a pair. You know that if you saw the film. Lawrence, whose name is on many things around the campus, was an experimentalist in nuclear physics. And for historical context it's worth saying that the neutron, one of the two basic nuclear particles, was only discovered in 1932. Within a year or two it had become a kind of probe experimentally into the nucleus, and within seven years it had been deployed to discover nuclear fission. And the story of nuclear fission and of nuclear weapons unfolds from there so fast in time. So let's round this out before we start our panel by saying something about the style of work and the form of life that Oppenheimer unfolded here at Berkeley in the 1930s especially. His scientific work was collaborative, pointed, directed at the hard problems, not always successful. And he landed in the bay at a time when much else was in ferment here, at the same time as he was devoting himself to theoretical physics and his students, he got engaged with contemporary left-wing politics in the bay area in the 1930s that included the fight against fascism in Nazi Germany and Spain, and also struggles for economic justice and labor in California. And as you probably know if you've seen the film, the Communist Party was part of that setting and he would pay for that during the McCarthy years after World War II. So Oppenheimer during his time at Berkeley immersed himself in the culture of the bay area, the life of the Berkeley faculty, intellectual currents across the university, this broad liberal arts institution that fed his roving mind. So with that as background, I want to start with some questions to my fellow panelists here. Let's begin with a question about science. This might be a question to start with Carl, I think, given your background. I've mentioned Ernest Lawrence as well as Robert Oppenheimer. And they were here at Berkeley together. How did they interact these two twin figures? Thank you, Catherine. This is a very interesting question. And so much of this early history is a story about just amazing timing and alignment of the stars, the fact that you had not just one, but these two figures appearing on the scene at the same time in Berkeley in the 1920s. Just to complete I think what Catherine sort of intimated, the U.S. at that time, both in theory and in experiment, was a backwater. There were some singular lights in both cases, but Europe was where all the developments were happening, the special relativity, the general relativity, the breaking of quantum mechanics in 1925, all in Europe. And you had here a remarkable thing happen in just a space of a few years in the late 1920s. First was Lawrence, who was purely an American phenomenon, very, very different from Oppenheimer. I think this is what makes the story quite interesting. Grew up in South Dakota, sort of good sturdy Lutheran stock, parents were school teachers, got his degrees at South Dakota in Minnesota, and did his PhD at Yale in 1925. And at this point it was recognized that he was going to be the leading light in the U.S. experimentally. He was all of a sudden a really hot commodity, got a National Research Council Fellowship, but he did not use it as normally people would do to travel to Europe. He stayed at Yale, and then all of a sudden was very much, many universities were contending for him, but he actually came out to UC Berkeley and became a faculty member here, I think in 1928. And he was already beginning to have ideas, the gestation of the cyclotron. It's hard to pin down exactly, but between 1929 and the conceptualization of it in 1932, this was a tool which was really going to revolutionize nuclear science. He was a very sort of a steamy person, very proper, I think it was reputed, he never was heard to say a foul word in his entire life, although he was a person of a very strong personality. And began immediately with the invention of the cyclotron, these grand visions of building, and as he executed bigger and bigger and bigger cyclotrons, the first one fit in the palm of your hand, it was invented literally 200 yards away in the physics building on the fourth floor. And all of a sudden there was an 11 inch, a 27 inch, a 37 inch, 60 inch and 184 inch cyclotron. If you drove up University Avenue, UC, on the hill there, this curious orange dome, they kept the original building, it's a historical building, there's some modern light source in there now. That was this 45,000 ton cyclotron, which was the largest in the world at that time. Then you have Oppenheimer, who came, went to Europe, got his degree at Goettingen, 1927 under Max Born, he brought the new physics back to the U.S. And again he was just on a trajectory, he was the leading U.S. theoretical physicist at the time. He was given this kind of jewel professorship, something I don't think would be done today, he was allowed to split his time between Berkeley and Caltech. And I think he arrived here in 29, I think Lawrence's presence must have been part of the attraction for him here. And they made an interesting pair, you had a person who was already very operating in the spirit of corporate physics, organizing a well run laboratory with grand divisions. And this Bohemian, this came from a sort of upper liberal Jewish family in New York City. One photograph that is blazing in my mind, it says it all, it comes from I think the 30s when they would hang out together at Oppenheimer's, where they went, the Oppenheimer family went for summers in the mountains of New Mexico. It shows the two of them in the same camera frame. And there's this very tall, spare looking Lawrence, very well groomed with spats.  The omnipresent cigarette is not looking at the camera. And you realize that this was kind of the odd couple. And they really worked very well together. And I think part of the attraction was their differences. I think even of the six children of Lawrence, one of them I think he named in Robert's honor. The beginning of the fissure began to be evident. The one thing that Lawrence was increasingly annoyed with early on was his Oppenheimer's left wanderings. He had none of this business of people trying to unionize the people in his laboratory and warned people, and particularly Oppenheimer, to leave politics aside and to focus on science. Which of course, that didn't happen. And that set up the story to evolve from there. But anyway, let me pitch it back to Catherine and Yasinoro. That's a wonderful grounding for actually, I think, a question for you, which is, what do you see as the key contributions that Oppenheimer made in science during this period? Were they solely tied to being here at Berkeley with Lawrence, or did he have his own set of questions to pursue? Yeah, it's kind of difficult. In fact, I'm kind of, I feel like a bit mispraced in the sense that if you see the movie, this movie is not about really literary physics.  And I give you some sense of what he did while he was at Berkeley. That was Oppenheimer-Sneider thing. That's the collaboration with the person called Sneider. That was a big thing. I will talk about it. But then I did it in a previous panel, and then one person sent me a long email, no, Oppenheimer's work is born Oppenheimer. So let me start from, born Oppenheimer is probably one of the most famous things about his work. And that was done when he was still in Europe. So that's the beginning of the quantum mechanics. Quantum mechanics was weird theory, like some single particle can exist multiple locations at the same time in parallel worlds or something. That's the one way of phrasing it. In fact, you have precise math, you can compute the probability of these particles floating around in many places, but you can't even phrase that thing. The people were really having trouble of understanding these things. This big monster, it's called quantum mechanics, was new at that time. And to test that idea, that theory with this experiment, and we needed to compute the process to compare with the experiment, one of the standard way of doing computation in a very complicated system was developed by Max Born in Germany and Oppenheimer. It's called Born Oppenheimer approximation of the thing. That was a major thing. But it's interesting that he's so famous, but his work was not like, to be honest, like world changing, like Albert Einstein, or Niez Bohr around that time, if you have heard the name, Werner Heisenberg. And of course, it's a top-rated physicist and top-rated work. Probably his strengths was rather like teaming up the people and really building a group and really starting the project which seemed impossible and make it really, that's including Manhattan Project or even building the quantum group in US that was here, of course. But Born Oppenheimer was probably the best known at that time when he came to Berkeley. And in movie, it's very difficult to explain because this is technical, like way to do approximation of a quantum mechanics zone. In movie, it's one word, molecules. As you said, like, oh, I know you work with molecules. This was Born Oppenheimer, so I wouldn't repeat it because otherwise it's dangerous. I probably keep talking at 58 hours and you probably won't be able to go back. So I will stop it here, but that's the key contribution when he came back to US, came to Berkeley. But here, he did something very interesting. To me, it's very, very, very interesting. It's affecting my research even now. That's called Oppenheimer and Snyder and also some other person. It's something called Black Hole. That's what he essentially discovered. The solution described in Black Hole was there before. It's called Carl Schwarzschild in Germany, it's 19, right after Einstein wrote down the general activity. But the meaning of that solution was not clear. He showed that when the star used up all nuclear energies, the star is bound using nuclear energy. Of course, nuclear physics was also new, but then the way star bonds is that if matter cramps and then, of course, matter attracted with each other by gravity. So if nothing happens, if no nuclear or anything, then the star must collapse because matter attracts. Gravity is universally attractive, no repulsion. So it's an attraction, attraction, attraction, it's becoming collapsing, collapsing, collapsing, smaller and smaller and smaller. But usually that's not what happens in sun, for example, because the sun wants to collapse, but at some point it's a collapse, and density becomes so high and the temperature becomes so hot, so the nuclear reactions start happening. And then energy was liberated, and this nuclear energy have a pressure, and that stops collapsing. And if nuclear energy was used up, and usually what happens is that now the sun starts contracting again, but that contraction is again stopped by some other thing called electron pressure, degeneracy pressure, and usually something's stopping collapsing. It doesn't happen to really to go to every material, go to the point. That doesn't happen. We don't see it. The chair is not collapsing to a point, and the desk is not collapsing to a point. But he said that in certain circumstances, suppose the original star was so big massive, and then he used up all the materials, energies and nuclear energies, and degeneracy pressure is not enough to stop, it's possible that all the material is really, really, really collapsing, nothing to stop, just the gravity make it collapse into a point. All the material, big material, composing entire planet, sun and so on, stars, it's really collapsing to a point. And he really computed that that can happen theoretically, and if that happens, the density is so high that nothing, gravity is so strong that nothing can escape from certain region around that point. So we cannot in principle even see inside, because even light cannot come to you. Light try to escape to the point, and then come back. Nobody can see certain region. It's essentially a totally different region from the rest of the world. It's called Black Hole right now, and in fact, it's observationally discovered only, I think, in this century. So that was really so strange object, the result of general relativity. That was not necessarily direct, I think, interaction with Ernest Lawrence, because it's so theoretical, but so he's doing his own thing here as well, and then it's very imaginary and very original, and it's even affecting my research, and that's the one thing I would pick, and I picked in a previous panel, and then somebody sent me a message that that was wrong pick, but anyway, I just repeat that anyway. And then that's the one he did here. So next is a question for both of you. Raise your hand. Would you have been surprised that Oppenheimer was then chosen to lead Los Alamos, given what you know about his background, his work style? Is it a surprising thing that that was who General Groves lit on to lead the project on the mesa of Los Alamos? I want to hear from both of you, and then I'll add some thoughts after I hear from you. Who goes first? I don't know much, but maybe I could go up. Okay. Just to my own background as an experimental nuclear physicist, I've been a junkie on all the literature of the Manhattan Project, and there's so much good literature out there on Oppenheimer, and the larger troika of Oppenheimer Lawrence and Edward Teller, who plays an interesting role in this overall story as well. Even to this day, and of course, the person whose call this was was General Leslie Groves, who I think is an overlooked figure in the overall Manhattan Project. He was, at the time, I think he was appointed as a Brigadier General just when he was appointed to run the Manhattan Project, of which Los Alamos was a very singular part, but by no means the whole thing. The overall Manhattan Project was a staggering enterprise where the US government literally opened up its checkbook without any limitation, and in the space of 30 or 32 months built up these incredibly massive facilities at Oak Ridge, Tennessee, for the separation of uranium, gaseous diffusion plants, the calutrons, the electromagnetic separation plants. These were buildings a mile long, and they were created almost overnight, and then there were actors at Hanford and Washington for creating plutonium. Groves was an Army Corps of Engineers. His previous accomplishment was he was the person that built the Pentagon, which may still be the world's largest building in terms of square footage. He was a master builder and a great planner and project manager, but I think he also understood people in a way that people did not appreciate. Here was Oppenheimer, who had a very tainted past. Even historians to this day argue whether or not he was a communist or not. This is still a matter of contention. His brother Frank certainly was, who I had actually had the opportunity to meet many years ago. He was the person who created the exploratory in San Francisco. His wife was a communist.  He took part in... Whether or not he was a full member or not is unclear, but he was a problematic character, and yet Groves recognized in him the ability to grasp all aspects of this very complex multi-physics, multidisciplinary problem of creating an atomic bomb. He also began to emerge, I think, this kind of magnetism personality-wise that could bring all these people that were fleeing from Europe over to work with him. On the other hand, you could have said, well, Lawrence was already a person that showed a kind of a masterful ability to organize. He was sort of the consummate corporate person. Arthur Compton, who ran the metallurgical laboratory in Chicago, another part of the Manhattan Project, was a person that many people said should have been the director of the project. In the end, it was the right decision, but it was... Even to this day, you could sort of say, what did Groves see at the time, and how was he able to see past all the problems to say, this is the person who uniquely should be... To me, it's a bit of a mystery, but go ahead. Me? Okay. All right. I don't know. I'm stealing the phrase from moving, but which I watched finally last night, but... No, it was cool. It was really, if you haven't seen it, you watched it.  It's very great. This is like building Secret City to work on this project, because it's a wartime project, and so, you know, Secret Things, Secret City, building from zero in the ranch, Neo Ranch, I guess, and New Mexico, and nothing's there. The building, all the city, just for scientists, bringing all the scientists all over the world, not from Germany, of course, and then he's a boss, and so, you know, it's a theoretical thesis, as you said, curious, weird people, set of weird people, you know, he must have been excited with that. I mean, and then he was the possibility of, say, being the director of that, right? He must have offered at some point, and then, yeah, I feel like he just said, yes, I mean, no other choice. I mean, later in life, of course, he must have contemplated the consequence of this bomb and so on, and so on, but at that point, my guess, simple guess, maybe wrong, but my simple guess is, wow, that's exciting, yes, just do it. That's, you know, the idea came from where I don't, I'm not really, that's probably communication between him and Leslie and then many people, but when it becomes possibility, I think he must have pushed like crazy. That's my guess. One thing that really came out in between both of your comments is also, Carl said it's a multi-physics problem. Designing a nuclear weapon is not going and taking existing blueprints and going and building. It's actually many different kinds of physics and chemistry that are involved that are cutting edge. If you think about building, say, the plutonium weapon, that's designing and constructing a weapon out of an element that did not exist on this earth in any measurable quantity before the early 1940s, discovered here at Berkeley too. So all of this was cutting edge science in secrecy, in the context of the incredible pressures of the Second World War. And so the challenge of being able to see across all of those disciplines and somehow pick out the crucial problem that needs to be solved at each point in time and bring the group together around it, thinking of it only at this point as a technical challenge racing the clock, because you remember probably, if you've seen the film, that the original concern that motivated the secret crash project and the bomb laboratory at Los Alamos was the concern that Nazi Germany would figure out before the United States and its allies how to build a nuclear weapon first. So it was a completely open question. The lines of communication were shut down. The refugees from Nazi Germany who came because they were expelled or saw the promise here in the United States were motivated hugely by the threat of what was going on in the Third Reich. So all of that pressure comes together at Los Alamos on a scientific problem whose solution is unclear. All that's known is that the clock is ticking. So the project has its own internal dynamics that Oppenheimer also becomes a crucial piece in managing, particularly as you get closer and closer to understanding the true feasibility, especially of the uranium weapon by roughly 1944 at the same time that it becomes clear as you as the Allied forces cross into Germany, that there is no German nuclear weapons project of any significance. And so at that point, the laboratory pivots and it's no longer a weapon that's imagined to be dropped on Berlin. Everyone continues working with very few exceptions because Oppenheimer is their charismatic leader. And then it pivots and points to the war in the Pacific and towards its use on Japan. So Oppenheimer is the motivator and the charismatic leader through the scientific and the political concerns at that point. And he keeps all of the scientists on the project moving forward with the results that we all know. So one of the things I think to come back to Carl's original point about Leslie Groves is yes, certainly Groves recognized the intellectual caliber and the leadership capacity, the leadership charisma of this man. But he also recognized that Oppenheimer was vulnerable because of his political background. And one of the things that as a historian I learned through the film was just the intelligence with which Groves understood that he could keep Oppenheimer under his thumb if needed, that I think the combination of Oppenheimer's ambition, the dynamics of his marriage, and the aspiration to do something world transformative sort of melded in his mind. And I think Groves understood that in him. You remember, Lawrence got his Nobel Prize at the end of the 1930s. Oppenheimer didn't. So what would be his legacy? In some sense, he made a choice. So with that, I think I want to segue a little bit towards the implications of scientific progress and the responsibilities of scientists. Question that the film raises very, very prominently and one that scientists have struggled with from the Los Alamos time forward. Do you think that Oppenheimer is a useful figure to think with? When imagining how scientific progress can lead towards outcomes that were completely unenvisioned. And how should a scientist bring that into their own thinking about their life's career? Would you like to? Yeah, I think that question is pretty relevant. It's actually serious one, right? I mean, even nowadays. So how much scientists should be responsible, or at least should be committed after certain technology or something was developed? It's not obvious. Of course, a final decision must come from politicians in a good sense, namely the people who are kind of selected by people at large. But then of course, scientists cannot say, in my opinion, it's like, okay, this I kind of developed this atomic bomb or something. I don't know how this is used. I mean, it's not my stuff. And that's one possible way of thinking, but I don't think, right, my personal opinion. And then that's what the story of Oppenheimer is giving us still. It's probably not maybe not the same magnitude in a sense. As you said, that that project was really unique, right? The project of science, big project of science, the power, nuclear power, human even didn't know that that kind of thing exists, like 10, 20 years before the project, maybe 30 years. And this is a project of the science, the project of the military thing, and then a huge political thing. It's all done in secret at the time of World War II. It's probably not the same level, even if it's quantum information or something which is related with national security nowadays. It's very important. But it's or cloning technology like CRISPR-Cas9 or something, we can change the genes to make up your own, you know, wanted child or something. What have to think about it is a very important subtle question. And scientists have to still be thinking about may not be the exactly same level of the thing like atomic bomb. But I think it's a very important thing. And nowadays, the story of Oppenheimer is extremely relevant for scientists and politicians, to everybody, I would say. If I can just riff on that a little bit. I think on the main scientists are people who do understand that they're creating technologies which are very often dual use. They can be for the good of humanity, for revolutions in medicine or what not, but can also be used for nefarious purposes as well. And many examples of this. I was a graduate student at MIT in the early 70s when recombinant DNA was developed. And I remember David Baltimore then at MIT, basically as a leading figure of the field, Nobel laureate, convened a moratorium for a while, getting the biologists together to discuss the implications of this. The same is happening today. A lot of the leading figures in AI are beginning to say, look, we need to tap the brakes, talk and advise the government here. On the main, though, that kind of conscience exerted by scientists and engineers sometimes is not by itself quite enough because there is such momentum when a technology gets rolling. It's like a big train and it's very, very difficult to stop. And I think as Freeman Dyson pointed out, his commentary on the Manhattan Project, at the time of VE Day, no one stopped working. I mean, they had a celebration. They just kept right on full bore. He said because the technology had so gripped them that it was at that time unstoppable. And then when it came time for the decision about the use, Dyson's comment was he said the president would have had to have been a man of steel to have said no at that point because the thing was moving with such incredible momentum, it was probably unstoppable. Maybe Catherine, you should throw in your own thoughts about this. My co-panelists know that I started out in theoretical physics in the 1980s, which was the end of the Reagan era before we knew that the Cold War was ending. Yeah. And at that point, the, the political, moral and ethical questions of what use young people's scientific talent should be used was front and center as we looked at the challenges around Star Wars around nuclear disarmament. And I remember at that time, personally as growing up in Washington, DC, thinking I was being coached, being trained to go into theoretical physics. And I expected sort of realized at the end of the 80s that that meant the path was open to the weapons laboratories. And it became very strangely personal at that point. It was actually one of the reasons that I chose in the early 90s to move in a different direction and became a historian of physics because these questions are very individual and personal. And at the same time, they play out against the backdrop of these world historical forces that none of us can necessarily shape or sometimes even understand. Knowing whether your scientific contributions will be used overall for the betterment of humanity, or for the creation of new catastrophes is, I think pretty much beyond any individual to say. So the place where this comes back from me now is, I now teach a course on the human context and ethics of data science and AI in Berkeley's amazing data science major, and walking the students through their own life choices, knowing that their own scientific work may escape their grasp. And yet, there's so so many of them are so fundamentally asking themselves, how can I do good with AI with machine learning with data science and to give them the chance to think through the experience of nuclear physicists at Los Alamos, and having Oppenheimer as a kind of stellar figure to think through becomes one of the exemplars that they can mirror and reflect their own experiences against. So in that sense, it sort of comes full circle. It's a decision that each of us who takes on specific expertise and often technical skill comes to grips with at different points in our career. And one of things that is very clear to me to tell my students is, I'm not as your professor, the one to tell you what's the right decision or the wrong decision. That's yours to work through. But it's one of the things that anyone who moves, I think this might be a moment to open it up to questions from the so we have two folks walking around. So you guys are bringing up AI and ethics and things like that. How are you thinking about people talking about a singularity? And what do we do about maintaining freshness and making sure that our young people are ahead of the game on all of this so that the ethics are in place.  So did everyone hear the question? It's drawing that arc to contemporary concerns and in this case around AI, but I'm sure we could bring it back to CRISPR-Cas9 and other scientific developments. I don't think there is a way to ensure that the arc of scientific development will always bend towards good. I think that's something fundamental for everyone at work in this field and I think in particular for those who work in computing and AI disciplines where the work that you make, the tools that you build can be used far outside of laboratories where you have control over how it gets used. In that situation where there is no guarantee that all will turn out well, one of the things Berkeley has invested in is giving our students the tools to think about which directions they want to try to move. They can't control the outcomes and I think my concern is actually less about existential risk and the singularity and Skynet and much more about algorithmic decision-making tools that get used in ways that reinforce biases or the sort of removal of our own agency to shape our political discourse in an age of deep fakes. I think there's no way except moment by moment for those who have the technical capacities to think where am I making choices that could lead down a road that I don't want to encourage. They can't force other people away from it, but they can make choices of their own. Forces entirely beyond his control as a physicist. Also, we don't really know. In AI case, for example, the best scenario or something, one possible scenario is that AI is doing something for us.  That's the kind of like as a tool, AI as a tool, that's the best kind of thing. The other one is that AI starts self-producing themselves and then tries to have a human and then take over like a carbon-based life and so on. They are self-reproducing and 10,000 years later, did you know that life was carbon-based before? Come on, it's true. So, and then evolution was something random thing from radioactive. I can't be true. I mean, we just have to, and that's, there are like a variety and the same for atomic bomb like could wipe out the world and then in the end that was not happening. So, probably what we have to do is keep watching and we really have to keep thinking and to try to use things in the right direction and we don't really know beforehand if everything could have happened like atomic bomb could have wiped out the earth and so far, fortunately, that was not the case in our universe. Hi, I have a question here. The film is particularly interesting to me, not only because my connection with Berkeley. I'm alum and I have two daughters. They're both current students of Berkeley. Also, my brother was a nuclear physicist working for Los Alamos National Lab for 20 years. So, my question is around how Oppenheimer was sort of mistreated later in his life, right? So, I guess he was mistreated partly or if not mainly because he's alleged connection with communism. And my question is that from your position, do you see that kind of situation could happen in the future to to scientists? I mean with, you know, for example, if the United States have political conflict with certain countries and so on. Yeah, thanks for bringing that up. There's no question that there was a real, you know, travesty of justice done to Oppenheimer and I think the case is made. Of course, you know, Louis Strauss and others really went after him. There was, you know, I think enormous prosecutorial misconduct there at his hearing. The rules were not at all fair. The unspoken thing, of course, was that, you know, the powers at that time were really pushing forward with the super project, the development of the hydrogen bomb, the thermonuclear weapon, a thousand times more powerful than the atomic bomb. And at which point, although Oppenheimer initially supported the research on it, at some point he said, no, we should at this point make an agreement with the Russians not to do this, with the Soviets, we should not do this. And that view was very unpopular and people found him inconvenient and it was important to kind of take him out of the inner circle of advising the government. And that was really, I think, the context which most people would agree upon. Of course, he did, was not his own best friend. I mean, his lapses and failures with the truth by today's standards with, you know, the current Department of Energy security policies for who is granted a clearance and who is not, remarkably consistent with the way it was 50 years ago or 70 years ago now. But I think today we would, you know, there are processes in place which would prevent that kind of, you know, egregious, you know, misconduct, judicial misconduct which happened in his case. You may be interested to know, and I think many academics know and people certainly in our business know, a very important event last year. It was after actually decades of similar attempts. Tom Mason, the director of Los Alamos National Laboratory, led an effort, himself plus eight of his predecessors, the living directors of Los Alamos, wrote a letter to the Secretary of Energy, Jennifer Granholm, a Berkeley faculty member, asking that the decision, the 1954 decision be vacated.  You can't restore his clearance, but they wanted the record to be made clear that, you know, he had been the victim of this, you know, judicial travesty. And she ultimately agreed with it. And if you read the letter of the directors and read, you know, the secretarial memorandum, they're both extraordinary documents. They're both of two or three pages long laying out the case, making clear that it is unclear today whether or not he would be granted a clearance or not based on his, you know, confabulation, particularly in the Huck and Chevalier case, but making it very clear that there was lack of process here. And I think they did the right thing. So in a sense, there has been some restoration of that, what I thought was a terrible deed.  Lawrence did? Do you think that was political or was he not as brilliant as the others? I mean, many people in a circle got a Nobel Prize, correct? Of course, Nobel Prize is just one of the prizes. And then a prize has a character, like, for example, like Stephen Hawking was like our, the superstar of our time did not get Nobel Prize. That's because nobody was usually given to some people, some work, not like as a person, but some particular work which really made things advance the world in a way that it can be really tested experimentally, very well, to the level that no doubt this is correct. And then, of course, Black Hole was discovered way, way later after he died. And given that, I don't think it's, to me, it may be totally political, and it's totally understandable that he did not Openheimer did not get the Nobel Prize and Lawrence did get the Nobel Prize. It's not, I think it's reasonable, that's my opinion, because Lawrence think really changed the world, by the way. The way we do, say particle physics and nuclear physics, and this is called accelerator nowadays, and cyclotone is one type of it, but the accelerator is the things we're doing. The biggest accelerator we have in Europe right now, like 27 kilometers, something, it's a huge thing, it's a couple of tens of billions that are project, he really created that kind of field, from the, essentially virtually from zero. And those are the things we get the Nobel Prize. Yes, Openheimer's work, a lot of things, a brilliant, but I don't, yeah, who knows, right? I mean, it's somebody's mind, and like mind of the committee members may be affected by political thing, like if this is a guy who developed the bomb, it's possible, but I think it's not just political, it's at least reasonable that he did not get the one. That's my view of personal opinion. In general, how do you think scientists view J. Robert Oppenheimer? How do we view him? Well, I think on the main, scientists view him very favorably, I think, and I think physicists, it kind of, it almost, you know, falls into the adulation. I think we also, you know, history, particularly the whole history of the Manhattan Project, is, you know, the book that the film was made on, American Prometheus, I think is well chosen because you could not see any element of modern human history that falls more perfectly into the template of a Greek tragedy. He's a, you know, a Greek tragic figure. You know, it was the sort of meteoric rise, and then the downfall, the seeds of which he bore within himself in many ways. He is a very complex character. If you read Gregory Hurkins' book, The Brotherhood of the Bomb, or is it Professor UC Santa Cruz, where he talks about the relationship of Edward Teller, Oppenheimer, and Lawrence, he makes the case that all of them were extremely brilliant people, and the bringing together of three of them really made something incredible happen in, you know, the 1940s, but they were all, they all had egos, egotistical, they were vain, they were complex characters, they all had faults, and I think when you begin to delve seriously into the history, you have to see the whole person, the whole Oppenheimer. On the whole, of course, he was a great figure, and we learned so much from him, but, you know, when you really begin to scratch below the surface, like all human beings, he's a very complex, fallible person. Catherine is the best person, I think, to, or? Yeah, I think, the scientist views, I think it's a favorably, it's a pretty great scientist, genuinely, and also he did a really great achievement, and, but there are, like you say, scientists, a lot of different types of scientists, right? Like, one of the scientists, which surrounding me nowadays, like, theoretical physics, like, oh, this equation, I found this equation, that's great, and then some voice somewhere, oh, that's sad, by the way, this equation is, all right, I mean, this, really just cares science, and just as science, and that's, that's funny, you might think that's good, the scientists must be like that, the pure scientist, and they're just a science, but, and that's great, I also value those scientists, that's really great, I probably want to be one of those scientists, but then, if everybody's like that, it's dangerous, people can use how to come with science, and scientists don't care at all, and, and the scientists also have to go to the commission, and so on, and really see what's going on, and how that technology is used, and to follow up those things, and those scientists are also needed, and talking to politicians, talking to a general public, and Oppenheimer is of the latter type, and those persons, those people are needed, and he is a master of those things, he did, I think he did his best, and we, I think, view favorably. I'll add only one small piece, Oppenheimer, as you probably know, was a polymathic intellectual, that was a mantle that he bore also with a fair amount of pride, his, his capacity to read Sanskrit in the original, his occupation with poets like John Dunn, so this sense that a scientist is not narrowly confined to technical work, but instead ranges across the entire scope of human intellectual experience, is a model that many aspiring scientists may hold to, and is then increasingly under pressure as the actual work of a scientific career forces specialization, forces going in depth into one thing, and knowing it better than anyone else in the world. So it's a tension, and Oppenheimer came to embody that tension in a way it was enabled by his time at Berkeley, and his access to the great intellectual treasures of this place, but he is also a kind of inspiration of broad knowledge and political reflection, and in that sense I think, I don't want to speak from my other colleagues, but I see some of that range and breadth in them as well. We have one final question. Thank you. I kind of got the impression that Oppenheimer did not treat security seriously, and that maybe I'm combining this with Feynman's recollections of the Manhattan Project, but he almost had open ridicule towards security procedures. Would you agree with that? Let's see. Well, of course, Feynman himself was a rascal. He used to play games with the security people down there, because he knew how to crack safe, so he would actually open up safes where classified information was kept and would leave love notes to the security people overnight, and it drove them crazy, so you have to take anything that Feynman says with a grain of salt. What has come out by serious scholarship, and certainly after the opening of the KGB files in 1992? Catherine, I think, can speak much more authoritatively to this. When you look at the Soviet side, there was no evidence of any malfeasance in his side. He was certainly not an agent of the Soviets. They did try to recruit him. He rebuffed them at every phase. Whether or not he treated security loosely or not, you have to remember back at that time, the whole thing about classified information and what's called SRD these days, secret restricted data needs to be handled, they were all making it up. It wasn't that long ago, during certainly the time when I was a postdoc at Berkeley, down on campus, there were faculty members here, the old-timers, who still had classified information in safes in their office. Finally, the DOE finally said, okay, enough, that stuff is done over at Livermore. Please give us your safes back. The secret meeting where they actually had the conception of how an implosion weapon might work was done in the building over here. It was a predecessor of the physics building. It was kind of a wooden shack. So they did the best they could, but there were conversations happening on railroad trains, on an airplanes, that would not be allowed to happen today among colleagues. But I don't think, I don't get the impression that they did anything that flaunted the regulations at the time, but I think Catherine made no more than I do. I'm sure that I don't, but one thing I would only add in closing is that connecting across scientific disciplines can't be done in secret silos. And that was one of the things that Oppenheimer recognized and defended and finally achieved at Los Alamos, recognizing that there was actually going to be no way to solve the scientific problem without the ability for scientists to speak freely with each other across their lines of the project. So I think like with Carl, I would not call that a breach of security practices. I would call it intelligent management of a cutting edge project under the evolving security regime of the Second World War. So with that, I believe you would like us to call it close. This was a great presentation. Thank you so much. I don't know. I just, because I wanted to say this and it's a missed opportunity, but I think this, all these Oppenheimer stories, something was very encouraging to me. I mean, I'm now the member of the Berkeley Center for Theoretical Physics. Thank you for some of you who supported that group. And that is the group Oppenheimer formed. And 80 years later, I am directing the group. I am from Japan, where the country he bombed in. And I'm very encouraged by all those stories. I'm very optimistic about amazing. I was really like, I am now as the leader of this group and Hiroshima and Saki from Japan, I think. Yeah, I think. Well, you know, he's had just an elbow operation, but the best baseball player in America now is Japanese. Shohei Otani is the best since Babe Ruth, so anyway.