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Rectangular Hyperbola

A line APB ...

Question

A line $A P B$ of constant length meets the x-axis at $A$ and y-axis at $B$ . If $A P = b, P B = a$ and the line slides with its extremities on the coordinate axes, show that equation of the locus of the point $P$ is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

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Solution

If $P$ be the point $(h, k)$ on the line, then
$h = O Q = R P = a cos θ$ , $k = P Q = b sin θ$
Eliminating $θ$ , we get
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

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