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Slope of a Line

C: x2+y2=9, ...

Question

$C : x^{2} + y^{2} = 9$ , $E : \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ , $L : y = 2 x$

Equation of the diameter of the ellipse $E$ conjugate to the diameter represented by $L$ is :

9 x + 2 y = 0

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2 x + 9 y = 0

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4 x + 9 y = 0

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4 x - 9 y = 0

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Solution

The correct option is B $2 x + 9 y = 0$
Let $y = m x$ be the diameter conjugate to the diameter $L : y = 2 x$ of the ellipse $E$ then,
$2 m = - \frac{4}{9}$ $(∵ m m^{'} = - \frac{b^{2}}{a^{2}})$
$\Rightarrow m = - \frac{2}{9}$ and the equation of the conjugate diameter is
$y = - \frac{2}{9} x$ or $2 x + 9 y = 0$

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