O-Level Coordinate geometry in two dimensions: worked solution

Question

A circle has equation x²+y²−8x−4y+15=0. Find its radius and the coordinates of its centre.

A circle in the form (x − a)² + (y − b)² = r² has centre (a, b) and radius r, so rewrite the equation in that form by completing the square in x and in y. x² + y² − 8x − 4y + 15 = 0 Group the x terms together and the y terms together: (x² − 8x) + (y² − 4y) + 15 = 0 Complete the square for x: half of −8 is −4, and (x − 4)² = x² − 8x + 16, so x² − 8x = (x − 4)² − 16 Complete the square for y: half of −4 is −2, and (y − 2)² = y² − 4y + 4, so y² − 4y = (y − 2)² − 4 Substitute both back in: (x − 4)² − 16 + (y − 2)² − 4 + 15 = 0 (x − 4)² + (y − 2)² − 5 = 0 (x − 4)² + (y − 2)² = 5 Compare with (x − a)² + (y − b)² = r²: a = 4 and b = 2, so the centre is (4, 2) r² = 5, so the radius is √5 (about 2.24)