Find the condition for the following set of curves to intersect orthogonally.
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and $x y = c^{2}$

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Solution

Given equations are
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ......... $(1)$ and $x y = c^{2}$ ....... $(2)$

New by differentiating these two equations
$\frac{2 x}{a^{2}} - \frac{2 y}{b^{2}} \frac{d y}{d x} = 0$ and $y + x \frac{d y}{d x} = 0$

or, $m_{1} = \frac{d y}{d x} = \frac{b^{2}}{a^{2}} \frac{x}{y}$ and $m_{2} = \frac{d y}{d x} = \frac{- y}{x}$

Now to intersect orthogonally, $m_{1} m_{2} = - 1$

$∴ \frac{b^{2}}{a^{2}} \frac{x}{y} \times \frac{- y}{x} = - 1$

or, $\frac{b^{2}}{a^{2}} = 1$

or, $b^{2} = a^{2}$ is the required condition.

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