Standard Equation of a Circle

Let P\left( {x,y} \right) be any point of the circle as shown in the diagram, then by the definition of circle, the distance of point P\left( {x,y} \right) from C\left( {h,k} \right) must be equal to the radius of the circle r. i.e. \left| {CP} \right| = r.


As we know that distance formula from the analytic geometry as

\left|  d \right| = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} -  {y_1}} \right)}^2}}

Now we shall use this formula to establish the equation of circle, consider the points C\left( {h,k} \right) = \left( {{x_1},{y_1}}  \right) and P\left( {x,y} \right) =  \left( {{x_2},{y_2}} \right), now using distance formula for these two points as:

\left|  {CP} \right| = r = \sqrt {{{\left( {x - h} \right)}^2} + {{\left( {y - k}  \right)}^2}}

Squaring both sides, we have

\boxed{{{\left(  {x - h} \right)}^2} + {{\left( {y - k} \right)}^2} = {r^2}}

This is the equation of circle with centre \left( {h,k} \right) and radius r. This is called the equation of circle in standard form.

Note: If the centre of the circle is at origin \left(  {0,0} \right), then h = 0,\,k = 0, so the equation of circle takes the form

\boxed{{x^2}  + {y^2} = {r^2}}

This is the equation of the circle with radius r and the centre at the origin \left( {0,0} \right) in two dimensions XY-plane.

Example: Find the equation of a circle with centre \left(  {1, - 2} \right) and radius 7.
Solution: Form the given data of the example we have centre \left( {1, - 2}  \right) and radius 7. In this situation we use the standard form of equation of circle is

{\left(  {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}

By the given condition h  = 1,\,k =  - 2 and r = 7, so putting these values in the given equation of circle, we have

\begin{gathered} {\left( {x - 1} \right)^2} + {\left( {y -  \left( { - 2} \right)} \right)^2} = {\left( 7 \right)^2} \\ \Rightarrow {\left( {x - 1} \right)^2} +  {\left( {y + 2} \right)^2} = 49 \\ \Rightarrow {x^2} - 2x + 1 + {y^2} + 4y + 4  = 49 \\ \Rightarrow {x^2} + {y^2} - 2x + 4y - 44 = 0 \\ \end{gathered}

This is the required equation of the circle.