A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². Many problems ask for the centre, the radius, or the equation itself.
Standard and general form
The standard form shows the centre and radius directly. The general form x² + y² + 2gx + 2fy + c = 0 hides them; we recover the centre and radius by completing the square in x and in y.
Key idea
(x − a)² + (y − b)² = r²: centre (a, b), radius r. The radius through two given endpoints of a diameter is half their distance apart.
Completing the square
Group the x-terms and y-terms, complete each square, and move the constant across to read r².
Worked example
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0. Complete the square: (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25. Centre (3, −2), radius 5.
Distance and midpoint formulas connect circle problems to coordinate geometry, letting you find a radius from two points or test whether a point lies on, inside or outside a given circle.
Remember
- Halve the coefficient of x (and of y) when completing the square.
- A point is inside the circle if its distance from the centre is less than r.