Question

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?

• A. $5:4$
• B. $14:13$
• C. $2:1$
• D. $13:11$

Answer 1

Answer: (A)

$5:4$

Equation of tangents
$y^2=12x$
$\Rightarrow y=2x+\frac{3}{m}$
$\frac{x^2}{1}-\frac{y^2}{8}=1$
$\Rightarrow y=mx\pm \sqrt{m^2-8}$
Since they are common tangent
$\therefore \frac{3}{m}=\pm \sqrt{m^2-8}$
$m^4-8m^2-9=0$
$m=\pm 3$
$\therefore \begin{array}{c}y=3x+1\\ y=-3x-1\end{array}>P(-\frac{1}{3},0),\begin{array}{c}S=(3,0)\\ S^{\prime }=(-3,0)\end{array}$