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Geometry Level 5

Tangent is drawn at any point $(p,q)$ on the parabola $y^2=4ax$ . Tangents are drawn from any point on this tangent to the circle $x^2+y^2=a^2$ , such that the chords of contact pass through a fixed point $(r,s)$ . Then $p,q,r,s$ hold which of the given relations?

r^2q=4p^2s

r^2q=-4p^2s

rq^2=4ps^2

rq^2=-4ps^2

1 solution

Aakash Khandelwal
Jun 8, 2016

We know tangent to a conic is given by T=0 ,

i.e. Tangent to parabola at $(p,q)$ is

$2ax -yq=-2ap$ .

Substitute x and y in this equation by $(x_1 , y_1)$ ,

Where $(x_1 , y_1)$ are points where tangent

$2ax_1 -y_1 q=-2ap$ .

Call this as equation $1$ .

Now equation of chord of contact of circle is T=0,passing through $(r,s)$ and $(x_1 , y_1)$ .

Therefore

$rx_1 + sy_1=a^{2}$ . ……… $2$ .

Since $1$ and $2$ are identical (in $x_1$ and $y_1$ ,

$r/2a=-s/q=-a/2p$ .

Now let each of these ratios be $\xi$ .

Now we get

$a=-2p \xi$

Since $r= 2a \xi = -4p( \xi)^{2}$

And $s=-q \xi$ .

Eliminating $\xi$ we get

$\boxed{rq^{2}=-4ps^{2}}$ .

Must be a level 3 or 4 problem

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