Question

If $y=m_{1}x+c_1$ and $y=m_{2}x+c_2,m_1\ne m_2$ 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

• A. $3+4\sqrt{2}$
• B. $-5+6\sqrt{2}$
• C. $-4+3\sqrt{2}$
• D. $7+6\sqrt{2}$

Answer 1

Answer: (C)

$C_1:x^2+y^2=2$
$C_2:y^2=x$
Let tangent to parabola be $y=mx+\frac{1}{4m}$.
It is also a tangent of circle so distance from centre of circle $(0,0)$ will be $\sqrt{2}$.
$\left|\frac{\frac{1}{4m}}{\sqrt{1+m^2}}\right|=\sqrt{2}$
$\Rightarrow 1=32m^2+32m^4$
by solving
$m^2=\frac{3\sqrt{2}-4}{8},m^2=\frac{-3\sqrt{2}-4}{8}\text{ (rejected) }$
$m=\pm \sqrt{\frac{3\sqrt{2}-4}{8}}$
$so,8\left|m_1{m}_2\right|=3\sqrt{2}-4$