Ex. 4. Use Ex. 2 and 3 to obtain the conic of closest contact at the
origin with x — y + xy + x 3 = 0.
[For the curve y = x + x 2 + 2x 3 + 2x i + 2x 5 + .... Comparing the co-
efficients of x, x 2 , x 3 , x* in this and the expansion of Ex. 3 we have

m = 1, a—1, h = 1, 6 = -2.
Other methods are (i) to differentiate the equations of curve and conic
four times and to identify the values of

dy <Py cPy d*y
J ' dx' dx 2 ' dx 3 ' dx*
for curve and conic at the origin ; (ii) to find the lines joining the
intersections of curve and conic to the origin and to choose m, a, h, b so
that five of them are y = x. ]

Ex. 5. Find the conies of closest contact at the origin with

a 2 x + by 2 + y 3 = and x 3 + y s = 3 axy.
[b 3 {a?x + by 2 ) = (a i x + a 2 b 2 y)x and if = Sax, x 2 = Sat/.]

Ex. 6. Find the parabolas of closest contact at the origin with

= x + y + xy + x 3 and y = x + x 2 - 2x s.
[x + y=y 2 and y — x = (2x-y) 2 .]

Ex. 7. Two curves have linear branches touching the same tangent
at 0, one having p-pomt and the other g-point contact with the tangent,
where p^q. Show that they meet q times at 0.
