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,

\[\begin{gathered} \frac{{OM}}{{OP}} = \cos \theta \\ \Rightarrow OM = OP\cos \theta \,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \frac{{MP}}{{OP}} = \sin \theta \\ \Rightarrow MP = OP\sin \theta \,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right) \\ \end{gathered} \]

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

\[\begin{gathered} x = r\cos \theta \\ y = r\sin \theta \\ \end{gathered} \]

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,

\[\begin{gathered} x = 5\cos t\,\,\, – – – \left( i \right) \\ y = 5\sin t\,\,\, – – – \left( {ii} \right) \\ \end{gathered} \]

Now squaring and adding equation (i) and (ii), we get
\[\begin{gathered} {x^2} + {y^2} = {\left( {5\cos t} \right)^2} + {\left( {5\sin t} \right)^2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 25{\cos ^2}t + 25{\sin ^2}t \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 25\left( {{{\cos }^2}t + {{\sin }^2}t} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 25\left( 1 \right) \\ \end{gathered} \]
Hence $${x^2} + {y^2} = 25$$ is the required equation of the circle.