Question

A tangent line $L$ is drawn at the point $(2,-4)$ on the parabola $y^2=8x.$ If the line $L$ is also tangent to the circle $x^2+y^2=a$, then ' $a$ ' is equal to _____

Answer 1

Answer: 2

tangent of $y^2=8x$ is $y=mx+\frac{2}{m}$
$P(2,-4)\Rightarrow -4=2m+\frac{2}{m}$
$\Rightarrow m+\frac{1}{m}=-2\Rightarrow m=-1$
$\therefore$ tangent is $y=-x-2$
$\Rightarrow x+y+2=\mathrm{0...}$(1)
(1) is also tangent to $x^2+y^2=a$
So $\frac{2}{\sqrt{2}}=\sqrt{a}$
$\Rightarrow \sqrt{a}=\sqrt{2}$
$\Rightarrow a=2$