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

Equation of tangent at $(2,-4)$
$T=0$
$\Rightarrow y(-4)=4(x+2)$
$\Rightarrow -y=x+2$
$\Rightarrow x+y+2=0$
Length of perpendicular from centre $(0,0)$ is equal to radius
$\Rightarrow \frac{2}{\sqrt{1+1}}=\sqrt{a}\Rightarrow \frac{2}{\sqrt{2}}=\sqrt{a}\Rightarrow a=2$