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Question
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
Solution
The condition for the curves \[a x^2 + b y^2 = 1 \text { and }a' x^2 + b' y^2 = 1\] to intersect orthogonally is given below :
\[\frac{1}{a} - \frac{1}{b} = \frac{1}{a'} - \frac{1}{b'}\]
\[\text { So, the condition for the curves } \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and }\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 to \text { intersect orthogonally is }\]
\[\frac{1}{\frac{1}{a^2}} - \frac{1}{\frac{1}{b^2}} = \frac{1}{\frac{1}{A^2}} - \frac{1}{\frac{- 1}{B^2}}\]
\[ \Rightarrow a^2 - b^2 = A^2 + B^2\]
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