Locus of the poles of the tangents to the hyperbola $x^{2} - y^{2} = (39)^{2}$ with respect to the parabola $y^{2} = 156 x$ is the ellipse $4 x^{2} + y^{2} = k$ ,where $k$ is equal to

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Solution

Given equation of hyperbola is
$x^{2} - y^{2} = (39)^{2}$
Equation of tangent to hyperbola is
$x sec θ - y tan θ = 39$ ....(1)
Let the pole be $(h, k)$ .Then equation of polar is
$k y = 2 \times 39 (x + h)$
$78 x - k y = - 78 h$ ....(2)
From (1) and (2),
$\frac{sec θ}{78} = \frac{tan θ}{k} = \frac{39}{- 78 h}$
$\Rightarrow sec θ = - \frac{39}{h}, tan θ = - \frac{k}{2 h}$
Since, ${sec}^{2} θ - {tan}^{2} θ = 1$
$\Rightarrow \frac{(39)^{2}}{h^{2}} - \frac{k^{2}}{4 h^{2}} = 1$
$\Rightarrow 4 h^{2} + k^{2} = 6084$
which is the locus of $(h, k)$
Given locus is $4 x^{2} + y^{2} = k$
$\Rightarrow k = 6084$

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