Question

Absoulte value of $y$-intercept of the common tangent to the parabola $y^{2}=32 x$ and $x^{2}=108 y$ is

Answer 1

Tangent to $y^{2}=32 x$ is $y=m x+\frac{8}{m}$
and tangent to $x^{2}=$ $108 y$ is
$y=m x-27 m^{2}$
$\therefore \frac{8}{m}=-27 m^{2}$
$\therefore m^{3}=\frac{-8}{27}$
$\therefore m=\frac{-2}{m}$
$\therefore y$-intercept $=\frac{8}{m}=8\left(\frac{-3}{2}\right)=-12$