Question

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

Answer 1

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