"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 :"

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Accepted Answer

"

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

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

For it being tangent to circle.

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

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

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

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

"

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 :

Solution

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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 :

Solution

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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 :