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

Let the centre of the circle passing through A(6, -6), B(3, -7) and C(3, 3) be O(x, y).
OA = OB
⇒ √((x-6)²+(y+6)²) = √((x-3)²+(y+7)²)
⇒ √(x²+36-12x+y²+36+12y) = √(x²+9-6x+y²+49+14y)
Squaring both the sides
x²+36-12x+y²+36+12y = x²+9-6x+y²+49+14y
⇒ 72-12x+12y = 58-6x+14y
⇒ -6x-2y = 58-72
⇒ -6x-2y = -14
⇒ 3x+y = 7
⇒ y = 7-3x …(i)
Similarly, OA = OC [Radii od circle]
⇒ √((x-6)²+(y+6)²) = √((x-3)²+(y-3)²)
⇒ √(x²+36-12x+y²+36+12y+36+12y) = √(x²+9-6x+y²+9-6y)
Squaring both the sides
x²+36-12x+y²+36+12y = x²+9-6x+y²+9-6y
⇒ 72-12x+12y = 18-6x-6y
⇒ -6x+18y = 18-72
⇒ -6x+18y = -54
⇒ x-3y = 9
⇒ x-3(7-3x) = 9
⇒ x-21+9x = 9
10x = 30 ⇒ x = 3
Putting the value of x in equation (i), we get
y = 7-3(3) = -2
Hence, the centre of circle is O(3, -2).

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