Question

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

Answer 1

Answer: 12

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