Find the cirumcentre of the triangle whose vertices are (-2, -3), (-1, 0), (7,-6).

Let the centre of the circle be O(x, y). The vertices A(-2, -3), B(-1, 0) and C(7, -6) at equal distance from the centre.

⇒ OA = OB = OC = radius

⇒ (2 + x) ² + (3 + y) ² = (1 + x) ² + y ²
⇒ 4 + 4x + x ² + 9 + 6y + y ² = 1 + 2x + x ² + y ²
⇒ 4 + 4x + 9 + 6y = 1 + 2x
⇒ 2x + 6y = -12
⇒ x + 3y = -6 ………………………..(i)

⇒ (1 + x) ² + y ² = (7 - x) ² + (6 + y) ²
⇒ 1 + 2x + x ² + y ² = 49 - 14x + x ² + 36 + 12y + y ²
⇒ 1 + 2x = 49 - 14x + 36 + 12y
⇒ 16x – 12y = 84
⇒ 4x – 3y = 21 ………………………..(ii)
Adding (i) and (ii), we get
x + 3y + 4x – 3y = -6 + 21
⇒ 5x = 15
⇒ x = 3
Substituting x = 3 in (i)
⇒ 4× 3 – 3y = 21
⇒ 12 – 3y = 21
⇒ −3y = 9
⇒ y = −3
Therefore the centre of the circle is (3, -3).