The centre of the circle passing through the points (6, –6), (3, –7) and (3, 3) is(A) (3, 2) (B) (–3, –2)(C) (3, –2) (D) (–3, 2)

To find the center of the circle passing through the given points, we can use the concept of the circumcenter. The circumcenter is the point of intersection of the perpendicular bisectors of the sides of a triangle.

Step 1: Find the midpoint of the line segment connecting the points (6, -6) and (3, -7).

• The midpoint formula is given by: ( (x1 + x2) / 2, (y1 + y2) / 2 )
• Applying the formula, we get: ( (6 + 3) / 2, (-6 - 7) / 2 ) = (4.5, -6.5)

Step 2: Find the midpoint of the line segment connecting the points (3, -7) and (3, 3).

• The x-coordinate of the midpoint will be the same as the x-coordinate of the given points since they lie on a vertical line.
• The y-coordinate of the midpoint is the average of the y-coordinates of the given points.
• Applying the formula, we get: (3, (-7 + 3) / 2 ) = (3, -2)

Step 3: Find the equation of the perpendicular bisector of the line segment connecting the points (6, -6) and (3, -7).

• The slope of the line segment is given by: (y2 - y1) / (x2 - x1)
• Applying the formula, we get: (-7 - (-6)) / (3 - 6) = -1/3
• The negative reciprocal of the slope is 3.
• Using the midpoint (4.5, -6.5) and the slope 3, we can find the equation of the perpendicular bisector using the point-slope form: y - y1 = m(x - x1)
• Substituting the values, we get: y - (-6.5) = 3(x - 4.5)
• Simplifying the equation, we get: y + 6.5 = 3x - 13.5

Step 4: Find the equation of the perpendicular bisector of the line segment connecting the points (3, -7) and (3, 3).

• The x-coordinate of the midpoint is the same as the x-coordinate of the given points since they lie on a vertical line.
• The slope of the line segment is undefined since the denominator is 0.
• The equation of the perpendicular bisector is a vertical line passing through the midpoint (3, -2).

Step 5: Find the point of intersection of the two perpendicular bisectors.

• Solving the equations of the perpendicular bisectors, we get: y + 6.5 = 3x - 13.5 and x = 3
• Substituting x = 3 into the first equation, we get: y + 6.5 = 3(3) - 13.5
• Simplifying the equation, we get: y + 6.5 = 9 - 13.5
• Solving for y, we get: y = -2

Step 6: The point of intersection of the perpendicular bisectors is the center of the circle passing through the given points.

• Therefore, the center of the circle is (3, -2).

Hence, the correct answer is (C) (3, -2).