The standard form
A circle is the set of all points a fixed distance (the radius) from a fixed point (the centre). If the centre is (a, b) and the radius is r, every point (x, y) on the circle satisfies the distance formula.
Key idea
(x − a)2 + (y − b)2 = r2, with centre (a, b) and radius r.
Watch the signs carefully: (x − 3)2 + (y + 2)2 = 25 has centre (3, −2) and radius √25 = 5, because y + 2 is really y − (−2).
The general form
Expanding the standard form gives x2 + y2 + 2gx + 2fy + c = 0. From this the centre is (−g, −f) and the radius is √(g2 + f2 − c). To read off the centre, halve the coefficients of x and y and change their signs; complete the square if you prefer.
Worked example
Find the centre and radius of x2 + y2 − 6x + 4y − 12 = 0.
Here 2g = −6 so g = −3; 2f = 4 so f = 2; c = −12. Centre = (−g, −f) = (3, −2). Radius = √(g2 + f2 − c) = √(9 + 4 + 12) = √25 = 5.
Diameters
If the two ends of a diameter are known, the centre is their midpoint and the radius is half the distance between them.
Remember
- A positive radius means g2 + f2 − c > 0, otherwise there is no real circle.
- The coefficients of x2 and y2 must be equal (and there is no xy term).
- A point is inside the circle if its distance from the centre is less than r.