Equations of a Circle

Let C be the centre of a circle with coordinates (h,k). If radius of the circle is R , then equation of such a circle is given by (x-h)² + (y-k)² = R² . Here we will explore the changes in circle with the changes made in the values of h , k and R. 

• When the centre of the circle coincides with the origin i.e. h = k = 0 . Equation of the circle is x² + y² = R² 
• When the circle passes through the origin i.e. R² = h² + k² . Equation of the circle is (x-h)² + (y-k)² = h² + k² or x² + y² – 2hx – 2 ky = 0. 
• When the circle touches x – axis i.e. R = k. Equation of the circle is (x-h)² + (y-R)² = R2 or x² + y² – 2hx – 2Ry + h² = 0 
• When the circle touches y - axis i.e. R = h. Equation of the circle is (x-R)² + (y-k)² = R² or x² + y² – 2Rx – 2ky + k² = 0 
• When the circle touches both the axes i.e. R = h = k. Equation of the circle is (x-R)² + (y-R)² = R² or x² + y² – 2Rx – 2Ry + R² = 0 
• When the circle passes through the origin and centre lies on x-axis i.e. k = 0 and h = R. Equation of circle is (x-R)² + (y-0)² = R² or x² + y² – 2xR = 0 
• When the circle passes through the origin and centre lies on y-axis i.e. h = 0 and k = R. Equation of circle is (x-0)² + (y-R)² = R² or x² + y² – 2yR = 0 

You can enter the equations of above seven types in the input box at the near bottom of the following applet and see the changes in circle. This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com