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^…

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 y2 = x, then the value of $8|{m_1}{m_2}|$ is equal to :

Accepted Answer

Let tangent to ${y^2} = x$ be

$$y = mx + {1 \over {4m}}$$

For it being tangent to circle.

$$\left| {{{{1 \over 4}m} \over {\sqrt {1 + {m^2}} }}} \right| = \sqrt 2 $$

$$ \Rightarrow 32{m^4} + 32{m^2} - 1 = 0$$

$$ \Rightarrow {m^2} = {{ - 32 \pm \sqrt {{{(32)}^2} + 4(32)} } \over {64}}$$

$$ \Rightarrow 8{m_1}{m_2} =  - 4 + 3\sqrt 2 $$

Solutions · 1