O-Level Coordinate geometry in two dimensions: worked solution

Question

A circle C is given by x² + y² − 10x + 6y + 9 = 0. Work out its centre and radius.

x² + y² - 10x + 6y + 9 = 0 The centre and radius can be read straight off a circle written as (x - h)² + (y - k)² = r², which has centre (h, k) and radius r. So the job is to rewrite the given equation in that form by completing the square. Step 1: group the x-terms together and the y-terms together. (x² - 10x) + (y² + 6y) + 9 = 0 Step 2: complete the square on the x-terms. Halve the coefficient of x: half of -10 is -5. (x - 5)² expands to x² - 10x + 25, which is 25 too much, so subtract it back: x² - 10x = (x - 5)² - 25 Step 3: complete the square on the y-terms. Halve the coefficient of y: half of 6 is 3. (y + 3)² expands to y² + 6y + 9, which is 9 too much, so subtract it back: y² + 6y = (y + 3)² - 9 Step 4: put both back into the equation and tidy up the constants. (x - 5)² - 25 + (y + 3)² - 9 + 9 = 0 (x - 5)² + (y + 3)² - 25 = 0 (x - 5)² + (y + 3)² = 25 Step 5: compare with (x - h)² + (y - k)² = r². (x - 5)² gives h = 5. (y + 3)² is the same as (y - (-3))², so k = -3. r² = 25, so r = √25 = 5. Centre (5, -3), radius 5.