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.


parametric-circle-eq

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.