# Parametric Equations of a Circle

Draw a circle with centre at $O\left( {0,0} \right)$ and with a radius equal to $r$ which is the fixed distance from the centre of the circle. Now let $P\left( {x,y} \right)$ be any point of the circle as shown in the diagram. Draw a perpendicular from point $P\left( {x,y} \right)$ on the X-axis, meeting at the point $M$. Consider the triangle $OMP$ which is a right angle triangle where $OM$ is the base of the right triangle and $MP$ is the perpendicular of the triangle.

From the basic ratios of trigonometry,

Since $OM = x$, $MP = y$, $OP = r$, putting these values in equation (i) and (ii) we get the following equations:

These equations are the called the parametric equations of a circle.

Example: Show that the parametric equations $x = 5\cos t$ and $y = 5\sin t$ represent the equation of circle ${x^2} + {y^2} = 25$.

Solution: We have been given parametric equations,

Now squaring and adding equation (i) and (ii), we get

Hence ${x^2} + {y^2} = 25$ is the required equation of the circle.