If the line $y = m x + c$ touches $x^{2} - y^{2} = 1$ and $y^{2} = 4 x$ , then $m^{2}$ is equal to

cos \frac{π}{5}

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2 cos \frac{π}{5}

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cos \frac{π}{10}

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2 cos \frac{π}{10}

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Solution

The correct option is B $2 cos \frac{π}{5}$
Let the tangent to parabola be $y = m x + \frac{1}{m}$
solving with hyperbola
$x^{2} - {(m x + \frac{1}{m})}^{2} = 1$
$\Rightarrow x^{2} (1 - m^{2}) - 2 x - \frac{1}{m^{2}} - 1 = 0$
For this line to be tangent,
$D = 0$
$\Rightarrow 4 + 4 \frac{(1 - m^{4})}{m^{2}} = 0 \Rightarrow m^{4} - m^{2} - 1 = 0$
$\Rightarrow m^{2} = \frac{1 + \sqrt{5}}{2} = 2 c o s \frac{π}{5}$

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