Question

Find the coordinates of the centre of the circle passing through the points P(6,-6), Q(3,-7)and R(3,3).

Answer 1

Let centre of circle is O ≡ (a , b)
We know, distance between centre and point lies on circumference of circle is always constant e.g., radius.
∴ OP = OQ = OR
Use distance formula, if $(x_1,y_1)$ and $(x_2,y_2)$ are two points then, distance between them = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

∵ OP = OQ = OR
∴ OP² = OQ² = OR²
(a - 6)² + (b + 6)² = (a - 3)² + (b + 7)² =(a - 3)² + (b - 3)²

Now, (a - 6)² + (b + 6)² = (a - 3)² + (b + 7)²
⇒a² - 12a + 36 + b² + 12b + 36 = a² - 6a + 9 + b² + 14b + 49
⇒-6a + 36 - 9 - 2b + 36 - 49 = 0
⇒ -6a - 2b + 72 - 58 = 0
⇒-6a - 2b + 14 = 0
⇒3a + b = 7 ------(1)

Similarly, (a - 6)² + (b + 6)² = (a - 3)² + (b -3)²
(a - 6 - a + 3)(a - 6 + a - 3) = (b - 3 - b - 6)(b - 3 + b + 6)
⇒-3(2a - 9) = -9(2b + 3)
⇒2a - 9 = 6b + 9
⇒a - 3b = 9 ----(2)

From equations (1) and (2),
9a + a = 21 + 9
a = 3 and b = -2

Hence centre is (3,-2)