Q.The coordinates of a moving point $P$ are $\left(\dfrac{a}{2}\left(\csc\theta + \sin\theta\right),\ \dfrac{b}{2}\left(\csc\theta - \sin\theta\right)\right)$, where $\theta$ is a variable parameter. Show that the equation of the locus of $P$ is $b^2x^2 - a^2y^2 = a^2b^2$.

A point is not a thing but a place — a position located, with reference to a coordinate system, by a unique ordered pair $(x,y)$: $x$ is the (signed) distance from the $y$-axis and $y$ the (signed) distance from the $x$-axis. The locus of a moving point is the path traced out by it as it moves subject to a stated geometric condition; equivalently, an equation in $x,y$ has infinitely many real solution-pairs $(x,y)$, and the collection of all the points corresponding to those solutions is the locus of the equation. (The plural of locus is loci.)

Some standard loci: a point moving so as to stay equidistant from two fixed points $A,B$ traces the perpendicular bisector of $AB$; a point equidistant from two fixed lines traces the angle bisector; a point at a fixed distance $r$ from a fixed point $O$ traces a circle of radius $r$ centred at $O$. A point on the rim of a circle rolling along a straight line traces a cycloid.

Procedure for finding the equation of a locus.

1. Assign coordinates $(h,k)$ to the moving point $P$ whose locus is wanted.
2. Translate the given geometric condition(s) into equation(s) relating $h,k$ and any parameters/known quantities.
3. Eliminate the parameter(s), so that only $h,k$ and known constants remain.
4. Replace $h$ by $x$ and $k$ by $y$; the resulting equation in $x,y$ is the equation of the locus.