## Monday, June 18, 2012

### 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