Locus of the mid-point of the chord of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ which is a tangent to the circle $x^{2} + y^{2} = c^{2}$ is

Moderate

A

${(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}})}^{2} = c^{2} (\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}})$
B

${(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}})}^{2} = c^{2} (\frac{x^{2}}{a^{4}} - \frac{y^{2}}{b^{4}})$
C

$(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}) = c^{2} {(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}})}^{2}$
D

$(\frac{x^{2}}{a^{4}} - \frac{y^{2}}{b^{4}}) = c^{2} {(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}})}^{2}$

Solution

Let $(h, k)$ be the mid-point of the chord, then its equation is $\frac{h x}{a^{2}} - \frac{k y}{b^{2}} = \frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}}$
Since it touches the circle $x^{2} + y^{2} = c^{2}$
$\frac{\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}}}{\sqrt{{(\frac{h}{a^{2}})}^{2} + {(\frac{k}{b^{2}})}^{2}}} = \pm c$
$\Rightarrow {(\frac{h^{2}}{a^{2}} - \frac{k^{2}}{b^{2}})}^{2} = c^{2} (\frac{h^{2}}{a^{4}} + \frac{k^{2}}{b^{4}})$
Required locus is
${(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}})}^{2} = c^{2} (\frac{x^{2}}{a^{4}} + \frac{y^{2}}{b^{4}})$

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