Question

If $y = m_1x + c_1$and $y = m_2x + c_2$, $m_1≠m_2$ are two common tangents of circle $x_2 + y_2 = 2$ and parabola $y^2 = x$, then the value of $8|m_1m_2|$ 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)

$-4+3\sqrt 2$

Answer 2

Suppose, tangent to $y^2 = x$ be $y=mx+\frac {1}{4m}$
For tangent to circle,
$|\frac {\frac 14m}{\sqrt {1+m^2}}|=\sqrt 2$
$32m^4 \+ 32m^2 – 1 = 0$
According to the Sridharacharya formula,
$m_2=\frac {−32±\sqrt {(32)^2+4(32)}}{64}$
$8m_1m_2=−4+3\sqrt 2$

So, the correct option is (C): $−4+3\sqrt 2$