Question

If tangents to the parabola $y^2=4ax$ intersect the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at A and B, then find the locus of point of intersection of tangents at A and B.

Answer 1

Let P = (h, k) be the point of intersection of tangents at A & B

$\therefore$ Equation of chord of contact AB is $\frac{xh}{a^2}-\frac{yk}{b^2}=1$

Which touches the parabola

Equation of tangent to parabola $y^2=4axy=mx+\frac{a}{m}\Rightarrow mx-y=-\frac{a}{m}$

Equation (i) & (ii) as must be same

$\therefore \frac{m}{\left(\frac{h}{a^2}\right)}=\frac{-1}{(-\frac{k}{b^2})}=\frac{-\frac{a}{m}}{1}\Rightarrow m=\frac{h}{k}\frac{b^2}{a^2}$ & $m=-\frac{ak}{b^2}\therefore \frac{n{b}^2}{k{a}^2}=-\frac{ak}{b^2}\Rightarrow$ locus of P is $y^2=-\frac{b^4}{a^3}.x$