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If the curves $a x^{2} + b y^{2} = 1$ and $c x^{2} + d y^{2} = 1$ intersects orthogonally, then $\frac{b - a}{d - c} =___$

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$\frac{a + b}{c + d}$

$\frac{a}{c} . \frac{b}{d}$

answer is D.

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Detailed Solution

$\begin{array}{l} a x^{2} + b y^{2} = 1 - - - - (1) \\ c x^{2} + d y^{2} = 1 - - - - (2) \end{array}$

(1) and (2) are orthogonal

$\begin{array}{l} \frac{1}{a} - \frac{1}{b} = \frac{1}{c} - \frac{1}{d} \\ \frac{b - a}{a b} = \frac{d - c}{c d} \Rightarrow \frac{b - a}{d - c} = \frac{a b}{c d} \end{array}$

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