Q.Write the parametric equations of the circle $(x-3)^2+(y+4)^2=25$.

Instead of describing a circle by an equation relating $x$ and $y$, the parametric form describes every point on it using a single variable $\theta$ — the angle that the radius to that point makes with the positive $X$-axis. For the standard circle $x^2+y^2=r^2$, dropping a perpendicular from a point $P(x,y)$ on the circle to the $X$-axis creates a right triangle giving $\cos\theta=x/r$ and $\sin\theta=y/r$, i.e.

For a circle centred at $(h,k)$ this becomes $x=h+r\cos\theta$, $y=k+r\sin\theta$. Because it involves only one variable, the parametric form is often quicker to work with than the two-variable Cartesian equation — particularly when finding a tangent at a point identified by its angle, or when a problem is naturally phrased in terms of an angle (as with tangents from two different inclinations, Miscellaneous Exercise Q.26).