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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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#### Solution

Let O (*x*, *y*) be the centre of the circle. And let the points (6, −6), (3, −7), and (3, 3) be representing the points A, B, and C on the circumference of the circle.

`:.OA = sqrt((x-6)^2+(y+6)^2)`

`OB = sqrt((x-3)^2+(y+7)^2)`

`OC = sqrt((x-3)^2+(y-3)^2)`

However OA = OB (Radii of same circle)

`=>sqrt((x-6)^2+(y+6)^2)=sqrt((x-3)^2+(y+7)^2)`

=>x^{2}+36 - 12x + y^{2} + 36 + 12y = x^{2} + 9 -6x + y2 + 49 -14y

⇒ -6x + 2y + 14 = 0

⇒ 3x + y = 7 ....1

Similary OA = OC (Radii of same circle)

`=sqrt((x-6)^2+(y+6)^2) = sqrt((x-3)^2 + (y -3)^2)`

=x^{2} + 36 - 12x +y^{2} + 36 + 12y = x^{2} + 9 - 6x + y^{2} + 9 - 6y

⇒ -6x + 18y + 54 = 0

⇒ -3x + 9y = -27 .....(2)

On adding equation (1) and (2), we obtain

10*y* = −20

*y* = −2

From equation (1), we obtain

3*x* − 2 = 7

3*x* = 9

*x* = 3

Therefore, the centre of the circle is (3, −2).

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