Q. Let $P$ be the point of intersection of the common tangents to the parabola $y^2 = 12x$ and the hyperbola $8x^2 - y^2 = 8$. If S and $S'$ denote the foci of the hyperbola where $S$ lies on the positive x-axis then $P$ divides $SS'$ in a ratio :
Solution:
Equation of tangents
$y^{2} = 12x \Rightarrow y = 2x +\frac{3}{m} $
$ \frac{x^{2}}{1} - \frac{y^{2}}{8} = 1 \Rightarrow y =mx \pm \sqrt{m^{2} -8} $
$ \therefore \frac{3}{m} = \pm \sqrt{m^{2} -8} $
$ m^{4} -8m^{2} -9 = 0 $
$ m = \pm3 $
Or $ \frac{x^{2}}{1} - \frac{y^{2}}{8} = 1$
$ e = 3 $
$ae = 3 $
