Question

Find the locus of point whose chord of contact w.r.t to the parabola $y^2=4bx$ is the tangents of the parabola $y^2=4ax$

Answer 1

Equation of tangent to $y^2=4ax$ is $y=mx+\frac{a}{m}$.....(1)

Let it is chord of contact for parabola $y^2=4bx$ w.r.t the point $P(h,k)$

Therefore equation of chord of contact is

$yk=2b(x+h)$

$y=\frac{2b}{k}x+\frac{2bh}{k}$..........(2)

From (1) and (2)

$m=\frac{2b}{k}$ $\frac{a}{m}=\frac{2bh}{k}$

$\Rightarrow a=\frac{4b^{2}h}{k^2}$

locus of P is

$y=\frac{4b^2}{a}x$