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