Let be any point of the circle as shown in the diagram, then by the definition of circle, the distance of point from must be equal to the radius of the circle . i.e. .
As we know that distance formula from the analytic geometry as
Now we shall use this formula to establish the equation of circle, consider the points and , now using distance formula for these two points as:
Squaring both sides, we have
This is the equation of circle with centre and radius . This is called the equation of circle in standard form.
Note: If the centre of the circle is at origin , then , so the equation of circle takes the form
This is the equation of the circle with radius and the centre at the origin in two dimensions XY-plane.
Example: Find the equation of a circle with centre and radius .
Solution: Form the given data of the example we have centre and radius . In this situation we use the standard form of equation of circle is
By the given condition and , so putting these values in the given equation of circle, we have
This is the required equation of the circle.