find the coordinates of the centre of the circle passing through the points P(6,-6), Q(3,-7) & R(3,3).​

(3,- 2).

Justification:

Suppose O (h, k) is the centre of the circle passing through the points P, Q and R.

OP = OQ ...[Radii of the same circle]

∴ √{(h - 6)² + [k - (- 6)²} = √{(h - 3)² + [k - (- 7)²}

...[By distance formula]

Squaring both side

∴ (h – 6)² + (k + 6)² = (h – 3)² + (k + 7)²

∴ h² – 12h + 36 + k² + 12k + 36 = h² – 6h + 9 + k² + 14k + 49

∴ 6h + 2k = 14

∴ 3h + k = 7 …(i)[Dividing both sides by 2]

OP = OR …[Radii of the same circle]

∴ √{(h - 6)² + [k - (- 6)²} = √{(h - 3)² + [k - 3)²}

∴ (h – 6)² + (k + 6)² = (h – 3)² + (k – 3)²

∴ h² – 12h + 36 + k² + 12k + 36 = h² – 6h + 9 + k² – 6k + 9

∴ 6h – 18k = 54

∴ 3h – 9k = 27 …(ii)[Dividing both sides by 2]

Subtracting equation (ii) from (i), we get

3h + k - 3h + 9k = 7 - 27

∴ 10k = -20

∴ k = -20/10 = -2

Substituting the value of k in equation (i), we get 3h – 2 = 7

∴ 3h = 9

∴ h = 9/3 = 3

∴ The co-ordinates of the centre of the circle are (3, -2).

Note: See attached picture.

step-by-step explanation: