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