The equation of a circle

Why (x − a)² + (y − b)² = r² is just the distance formula squared, reading the centre and radius off it, and completing the square to recover them from the expanded form x² + y² + 2gx + 2fy + c = 0.

Pure · Coordinate geometry

The equation of a circle

Why it works

A circle is the set of all points a fixed distance

r

(the radius) from a fixed point

(a, b)

(the centre). Take any point

(x, y)

on it: the distance from the centre is

\sqrt{(x - a)^2 + (y - b)^2}

, and that must equal

r

. Square both sides and you have the equation of the circle:

(x - a)^2 + (y - b)^2 = r^2.

So it is literally the length formula with the square root undone. Read it carefully: the centre coordinates are the numbers that make each bracket zero, so

(x - 3)^2 + (y + 2)^2 = 16

has centre

(3, -2)

— the sign flips — and the right-hand side is

r^2 = 16

, so the radius is

\sqrt{16} = 4

, not

16

.

Examiners often hand you the circle multiplied out, in the form

x^2 + y^2 + 2gx + 2fy + c = 0.

You can't read the centre off this directly — you have to rebuild the brackets by completing the square in

x

and in

y

separately:

x^2 + 2gx = (x + g)^2 - g^2, \qquad y^2 + 2fy = (y + f)^2 - f^2.

Putting it back together gives

(x + g)^2 + (y + f)^2 = g^2 + f^2 - c

, so the centre is

(-g, -f)

and the radius is

\sqrt{g^2 + f^2 - c}

. (For this to be a real circle you need

g^2 + f^2 - c > 0

— otherwise the "radius squared" is negative and there are no points.) Don't trust the memorised formula blindly; completing the square on the spot is safer and is what the marks are for.

To find a circle's equation you need a centre and a radius. Two classic routes: the radius is the distance from the centre to any point on the circle; and if you're given the ends of a diameter, the centre is their midpoint and the radius is half the distance between them.