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 $S^{{}^{\prime }}$ denote the foci of the hyperbola where $S$ lies on the positive $x$ -axis then $P$ divides $S{S}^{{}^{\prime }}$ 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.
$\Rightarrow {\left(\frac{3}{m}\right)}^2=m^2-8$
$\Rightarrow 9=m^2\left(m^2-8\right)$
$\Rightarrow m^2=9\&m^2=-1$
$\Rightarrow m=\pm 3$
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 $a=1,b=2\sqrt{2},8=1\left(e^2-1\right)$
$\Rightarrow e=3$
$S\left(3,0\right)\&S^{{}^{\prime }}\left(-3,0\right)$
Let intersection point divides foci in $k:1$ ratio.
$\Rightarrow \frac{-1}{3}=\frac{-3k+3}{k+1}$
$\Rightarrow k=5:4.$