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)

Given equations
parabola $y^{2}=12x$
hyperbola $8x^{2}-y^{2}=8.$
Slope tangent forms to parabola and hyperbola are
$y=mx+\frac{3}{m}\&y=mx\pm\sqrt{1 \left(m\right)^{2} - 8}$
Above lines are representing same equation.
$\Rightarrow \left(\frac{3}{m}\right)^{2}=m^{2}-8$
$\Rightarrow 9=m^{2}\left(m^{2} - 8\right)$
$\Rightarrow m^{2}=9\&m^{2}=-1$
$\Rightarrow m=\pm3$
Common tangents are
$y=3x+1\&y=-3x-1.$
Point of intersection of above tangents is $P\left(\frac{- 1}{3} , 0\right)$
Foci of hyperbola $a=1,b=2\sqrt{2},8=1\left(e^{2} - 1\right)$
$\Rightarrow e=3$
$S\left(3 , 0\right)\&S^{'}\left(- 3 , 0\right)$
Let intersection point divides foci in $k:1$ ratio.
$\Rightarrow \frac{- 1}{3}=\frac{- 3 k + 3}{k + 1}$
$\Rightarrow k=5:4.$