a. The center of the circle is the midpoint of its diameter. The coordinates of the midpoint of a line segment are the average of the coordinates of the end points. If the center is point C, then
C = (P +Q)/2
C = ((-10, -2) +(4, 6))/2 = ((-10+4)/2, (-2+6)/2)
C = (-6/2, 4/2) = (-3, 2)
b. The radius is half the diameter, so is half the length of the segment PQ. It is also the distance from C to P or Q. The Pythagorean theorem is used to find this length.
|Q -C| = |(4, 6) -(-3, 2)| = |(7, 4)|
= √(7² +4²) = √65
c. The equation of a circle with center (h, k) and radius r is ...
(x -h)² +(y -k)² = r²
For your given center and radius, the equation of the circle is
(x +3)² +(y -2)² = 65
