If the chord of contact of tangents from a point `P` to the parabola `y^2=4a x` touches the parabola `x^2=4b y ,` then find the locus of `Pdot`

Options

A. A circle
B. A parbola
C. A pair of straight lines
D. A hyperbola

Answer

D

Explanation

`QR:y beta = 2a(x+alpha)`
`y=(2a)/(beta) (x+alpha)`
Parabola : `x^(2)=4by`
`rArr y=(2a)/(beta) (x+alpha)`
Parabola: `x^(2)=4by`
`rArr x^(2)=4b.(2a)/(beta) rArr beta x^(2) = 8ab (x+alpha) rArr Bx^(2)-8ax -8ab alpha = 0 " " :' " " D=0`
`rArr 64 a^(2)b^(2) +32 ab alpha beta = 0 rArr alpha beta = (64^(2)b^(2))/(32ab) = -2ab rArr "Locus of " (alpha, beta) ` is a hyperbola