Find the centre of a circle passing through t | vedclass

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(A) Let $O(x, y)$ be the centre of the circle. Let the points $A(6, -6), B(3, -7),$ and $C(3, 3)$ lie on the circumference.Since the distance from the

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$(3, -2)$

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(A) Let $O(x, y)$ be the centre of the circle. Let the points $A(6, -6), B(3, -7),$ and $C(3, 3)$ lie on the circumference.Since the distance from the centre to any point on the circle is equal (radius $r$):$OA^2 = OB^2 = OC^2$$OA^2 = (x - 6)^2 + (y + 6)^2$$OB^2 = (x - 3)^2 + (y + 7)^2$$OC^2 = (x - 3)^2 + (y - 3)^2$Equating $OB^2 = OC^2$:$(x - 3)^2 + (y + 7)^2 = (x - 3)^2 + (y - 3)^2$$(y + 7)^2 = (y - 3)^2$$y^2 + 14y + 49 = y^2 - 6y + 9$$20y = -40 \Rightarrow y = -2$Equating $OA^2 = OC^2$:$(x - 6)^2 + (y + 6)^2 = (x - 3)^2 + (y - 3)^2$Substitute $y = -2$:$(x - 6)^2 + (-2 + 6)^2 = (x - 3)^2 + (-2 - 3)^2$$(x - 6)^2 + 16 = (x - 3)^2 + 25$$x^2 - 12x + 36 + 16 = x^2 - 6x + 9 + 25$$-6x = -18 \Rightarrow x = 3$Thus,the centre is $(3, -2)$.

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

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