The coordinates of a moving point P are $\left(\dfrac{a}{2} \left(\text{cosec } \theta + \sin \theta\right), \dfrac{b}{2} \left(\text{cosec } \theta - \sin \theta\right)\right)$, where $\theta$ is a variable parameter. Find the equation of locus of P.
Let $P \left(h, k\right)$ be the moving point.
Then as per question,
$h = \dfrac{a}{2} \left(\text{cosec } \theta + \sin \theta\right)$, $k = \dfrac{b}{2} \left(\text{cosec } \theta - \sin \theta\right)$
i.e. $\dfrac{2h}{a} = \text{cosec } \theta + \sin \theta$ $\;\;\; \cdots (1)$
and $\dfrac{2k}{b} = \text{cosec } \theta - \sin \theta$ $\;\;\; \cdots \; (2)$
Squaring and subtracting equations $(1)$ and $(2)$ gives
$\dfrac{4h^2}{a^2} - \dfrac{4k^2}{b^2} = \left(\text{cosec } \theta + \sin \theta\right)^2 - \left(\text{cosec } \theta - \sin \theta\right)^2$
i.e. $4\left(\dfrac{h^2}{a^2} - \dfrac{k^2}{b^2}\right) = \text{cosec}^2 \theta + \sin^2 \theta + 2 \text{cosec } \theta \sin \theta - \text{cosec}^2 \theta - \sin^2 \theta + 2 \text{cosec } \theta \sin \theta$
i.e. $4\left(\dfrac{h^2}{a^2} - \dfrac{k^2}{b^2}\right) = 4 \text{cosec } \theta \sin \theta$
i.e. $\dfrac{h^2}{a^2} - \dfrac{k^2}{b^2} = 1$
i.e. $h^2 b^2 - k^2 a^2 = a^2 b^2$
$\therefore$ $\;$ The required equation of locus is: $\hspace{1em}$ $x^2 b^2 - y^2 a^2 = a^2 b^2$