Theory question 109
Coordinate GeometryA circle is drawn through the points (3, 2), (−1, −2) and (5, −4). Find the: (a) coordinates of the centre of the circle; (b) radius of the circle; (c) equation of the circle.
Show the model answer
Model answer
(a) Centre (h, k) is equidistant from the three points:
from (3, 2) and (−1, −2): h + k = 1; from (3, 2) and (5, −4): h − 3k = 7.
k = −3/2, h = 5/2: centre (5/2, −3/2).
(b) r² = (3 − 5/2)² + (2 + 3/2)² = 1/4 + 49/4 = 25/2, so r = 5/√2 ≈ 3.54.
(c) (x − 5/2)² + (y + 3/2)² = 25/2, i.e. x² + y² − 5x + 3y − 4 = 0.
Explanation
Equal distances from the centre give two linear equations.