Question

Show that the points (3, 4), (0, 5), (–3, –4) and
(–5, 0) are concyclic and find the radius of the
circle.

Answer 1

(x - h)² + (y - k)² = r² where (h, k) is the circle's centre and r is the radius.

Substitute the points:

h² + (5 - k)² = r² = (5 + h)² + k²

» h² + 25 - 10k + k² = 25 + 10h + h² + k²

» -10k = 10h, k = -h

(3 - h)² + (4 - k)² = (3 + h)² + (4 + k)²

» 9 - 6h + h² + 16 - 8k + k² = 9 + 6h + h² + 16 + 8k + k²

» -12h - 16k = 0

» 3h + 4k = 0

» k = -h

• So h and k are zero, therefore r = 5 (x² + y² = 25).