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Eccentricity of Hyperbola

The locus of ...

Question

The locus of poles w.r.t the parabola $y^{2} = 4 a x$ of tangents to the hyperbola $x^{2} - y^{2} = a^{2}$ is

x^{2} + 4 y^{2} = 8 a^{2}

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x^{2} + 4 y^{2} = 2 a^{2}

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4 x^{2} + y^{2} = 4 a^{2}

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2 x^{2} + y^{2} = 4 a^{2}

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Solution

The correct option is B $4 x^{2} + y^{2} = 4 a^{2}$
Given parabola $y^{2} = 4 a x$ and hyperbola $x^{2} - y^{2} = a^{2} \Rightarrow \frac{x^{2}}{a^{2}} - \frac{y^{2}}{a^{2}} = 1$
Polar of point P $(X, Y)$ wrt the given parabola is $y Y - 2 a x - 2 a X = 0$
Now the condition of tangency is $a^{2} l^{2} - b^{2} m^{2} = n^{2}$
Substitute values, we get
$\Rightarrow a^{2} {4 a}^{2} - a^{2} Y^{2} = {4 a^{2} X}^{2} \Rightarrow 4 a^{2} - Y^{2} = 4 X^{2} \Rightarrow 4 X^{2} + Y^{2} = 4 a^{2}$

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x^{2} - y^{2} = a^{2}

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