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Q.
If $y = m _{1} x + c _{1}$ and $y = m _{2} x + c _{2}, m _{1} \neq m _{2}$ are two common tangents of circle $x^{2}+y^{2}=2$ and parabola $y ^{2}= x$, then the value of $8\left| m _{1} m _{2}\right|$ is equal to
$C_{1}: x^{2}+y^{2}=2$
$C_{2}: y^{2}=x$
Let tangent to parabola be $y = mx +\frac{1}{4 m }$.
It is also a tangent of circle so distance from centre of circle $(0,0)$ will be $\sqrt{2}$.
$\left|\frac{\frac{1}{4 m }}{\sqrt{1+ m ^{2}}}\right|=\sqrt{2}$
$ \Rightarrow 1=32 m ^{2}+32 m ^{4}$
by solving
$m ^{2}=\frac{3 \sqrt{2}-4}{8}, m ^{2}=\frac{-3 \sqrt{2}-4}{8} \text { (rejected) }$
$m=\pm \sqrt{\frac{3 \sqrt{2}-4}{8}}$
$so , 8\left| m _{1} m _{2}\right|=3 \sqrt{2}-4$