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Find the centre of a circle passing through the points (6, - 6), (3, - 7) and (3, 3).

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Given that circle passes through the points A(6, - 6), B(3, - 7), C(3, 3).

Let us assume O(x, y) be the centre of the circle.

We know that distance from the centre to any point on h circle is equal.

So, OA = OB = OC

We know that distance between two points (x1, y1) and (x2, y2) is

Now,

⇒ OA = OB

⇒ OA2 = OB2

⇒ (x - 6)2 + (y - (- 6))2 = (x - 3)2 + (y - (- 7))2

⇒ (x - 6)2 + (y + 6)2 = (x - 3)2 + (y + 7)2

⇒ x2 - 12x + 36 + y2 + 12y + 36 = x2 - 6x + 9 + y2 + 14y + 49

⇒ 6x + 2y = 14

⇒ 3x + y = 7 ..... (1)

Now,

⇒ OB = OC

⇒ OB2 = OC2

⇒ (x - 3)2 + (y - (- 7))2 = (x - 3)2 + (y - 3)2

⇒ (x - 3)2 + (y + 7)2 = (x - 3)2 + (y - 3)2

⇒ x2 - 6x + 9 + y2 + 14y + 49 = x2 - 6x + 9 + y2 - 6y + 9

⇒ 20y = - 40

⇒ y = - 2 .... - (2)

Substituting (2) in (1), we get

⇒ x = 3

∴ (3, - 2) is the centre of the circle.

We know radius is the distance between the centre and any point on the circle.

Let ‘r’ be the radius of the circle.

⇒ r = √(9 + 16)

⇒ r = √25

⇒ r = 5

∴ The radius of the circle is 5.

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