Question

In what ratio, the point of intersection of the common tangents to hyperbola $\frac{x^2}{1}-\frac{y^2}{8}=1$ and parabola $y^2=12x$, divides the foci of the given hyperbola?

• A. $3:4$
• B. $3:2$
• C. $5:4$
• D. $5:3$

Answer 1

Answer: (C)

$5:4$

Let equation of common tangent is $y=mx+\frac{3}{m}$
$\therefore {\left(\frac{3}{m}\right)}^2=1.m^2-8$
$\Rightarrow m^4-8m^2-9=0$
$\Rightarrow m^2=9$
$\Rightarrow m=\pm 3$
$\therefore$ equation of common tangents are $y=3x+1$ & $y=-3x+1$
$\therefore \frac{PS}{P{S}^{\prime }}=\frac{3+\frac{1}{3}}{-\frac{1}{3}+3}=\frac{5}{4}$.