Which of the following gives the equation of director circle of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ ?

$x^{2} - y^{2} = 9$

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$x^{2} + y^{2} = 41$

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$x^{2} + y^{2} = 9$

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$x^{2} + y^{2} = 25$

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Solution

The correct option is B
$x^{2} + y^{2} = 41$

The equation of director circle of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 i s x^{2} + y^{2} = a^{2} + b^{2}$

$\Rightarrow D i r e c t o r c i r c l e o f \frac{x^{2}}{25} + \frac{y^{2}}{16} = 1 i s,$

$x^{2} + y^{2} = 41$

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