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Geometry Level 3

Locus of the points of intersection of two tangents to the parabola y 2 = 4 a x y^2=4ax at t t and 2 t 2t is

Note:- t t is the parametric point of the parabola i . e . i.e. ( a t 2 , 2 a t ) (at^2,2at)

3 y 2 = 4 a x 3y^2=4ax 2 y 2 = 9 a x 2y^2=9ax 4 y 2 = 9 a x 4y^2=9ax 3 y 2 = 8 a x 3y^2=8ax

Skanda Prasad by Skanda Prasad , 20, India

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1 solution

Skanda Prasad
Sep 22, 2017

Note that equation of the tangent in parametric form for the parabola y 2 = 4 a x y^2=4ax is given by y t − x = a t 2 yt-x=at^2

At t t and 2 t 2t the equation of the tangents are y t − x = a t 2 yt-x=at^2 and 2 y t − x = 4 a t 2 2yt-x=4at^2 .

Solving both the equations, we get ( x , y ) = ( 2 a t 2 , 3 a t ) (x,y)=(2at^2,3at) .

⟹ \implies t = y 3 a t=\dfrac{y}{3a}

⟹ \implies t 2 = y 2 9 a 2 t^2=\dfrac{y^2}{9a^2}

And t 2 = x 2 a t^2=\dfrac{x}{2a}

Therefore, x 2 a = y 2 9 a 2 \dfrac{x}{2a}=\dfrac{y^2}{9a^2}

⟹ \implies 2 y 2 = 9 a x 2y^2=9ax

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