Question

If the tangent to the parabola $y^2=4ax$ intersect the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at $P$ and $Q$ and the locus of the point of intersection of the tangents at $P$ and $Q$ is $y^{\alpha }=-\frac{b^{\beta }}{a^{\gamma }}x$, then $\alpha +\beta +\gamma$ is

Answer 1

Let $P(h,k)$ be point of intersection of the tangents at $P$ and $Q$,
Equation of the chord of contact is,
$\frac{hx}{a^2}-\frac{ky}{b^2}=1\Rightarrow y=\frac{b^{2}h}{a^{2}k}x-\frac{b^2}{k}\cdots \left(1\right)$
This touches the parabola,

The equation of tangent of a parabola is,
$y=mx+\frac{a}{m}\cdots \left(2\right)$
Comparing equation $\left(1\right)$ and $\left(2\right)$,
$m=\frac{b^{2}h}{a^{2}k}\frac{a}{m}=-\frac{b^2}{k}\Rightarrow a=-\frac{b^2}{k}\times \frac{b^{2}h}{a^{2}k}\Rightarrow k^2=-\frac{b^4}{a^3}h$
So, the locus is,
$y^2=-\frac{b^4}{a^3}x$
$\alpha +\beta +\gamma =9$