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The differential equation of rectangular hyperbolas whose axes are asymptotes of the hyperbola $x^{2} - y^{2} = a^{2}$ , is

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$x d y + y d x = C$

$x \frac{d y}{d x} = y$

$y \frac{d y}{d x} = x$

$x \frac{d y}{d x} = - y$

answer is B.

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Detailed Solution

The equation of the rectangular hyperbola whose axes are asymptotes of the hyperbola $x^{2} - y^{2} = a^{2}$ is $x y = c^{2}$

where $c^{2} = a^{2} / 2$ This is a one parameter family of curves. Differentiating with respect to x, we get

$x \frac{d y}{d x} + y = 0 \Rightarrow x \frac{d y}{d x} = - y$

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