Find the equation of a circle using the centre and radius
To find the equation of a circle when you know the radius and centre, use the formula \({(x - a)^2} + {(y - b)^2} = {r^2}\), where \((a,b)\) represents the centre of the circle, and \(r\) is the radius.
This equation is the same as the general equation of a circle, it's just written in a different form.
Example
Find the equation of the circle with centre \((2, - 3)\) and radius \(\sqrt 7\).
\({(x - 2)^2} + {(y - ( - 3))^2} = {\left( {\sqrt 7 } \right)^2}\)
\({(x - 2)^2} + {(y + 3)^2} = 7\)
If required for further work you can expand this to give:
\({x^2} - 4x + 4 + {y^2} + 6y + 9 - 7 = 0\)
\({x^2} + {y^2} - 4x + 6y + 6 = 0\)
Question
Find the equation for the circle with centre \(= (1,2)\) and radius \(= \sqrt 5\)
\({(x - 1)^2} + {(y - 2)^2} = {\left( {\sqrt 5 } \right)^2}\)
\({x^2} - 2x + 1 + {y^2} - 4y + 4 = 5\)
\({x^2} + {y^2} - 2x - 4y = 0\)
Question
Find the equation for the circle with centre \(= (0,0)\) and radius \(= 4\)
\({(x - 0)^2} + {(y - 0)^2} = {\left( 4 \right)^2}\)
\({x^2} + {y^2} - 16 = 0\)
