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

The Correct Option is C

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): \(−4+3\sqrt 2\)