A diameter intersects a circle at the points (6,−4) and (−2, 6).(a) Find the centre and radius of the circle.

The equation of a circle is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius.

Given two points on the circle (x1, y1) = (6, -4) and (x2, y2) = (-2, 6), we can find the center of the circle, which is the midpoint of the diameter. The midpoint (h, k) is given by:

h = (x1 + x2) / 2 k = (y1 + y2) / 2

Substituting the given points:

h = (6 - 2) / 2 = 2 k = (-4 + 6) / 2 = 1

So, the center of the circle is (2, 1).

Next, we find the radius of the circle. The radius is the distance from the center of the circle to any point on the circle. We can use the distance formula:

r = sqrt[(x - h)² + (y - k)²]

Substituting the center (h, k) = (2, 1) and one of the points on the circle (x, y) = (6, -4):

r = sqrt[(6 - 2)² + (-4 - 1)²] r = sqrt[(4)² + (-5)²] r = sqrt[16 + 25] r = sqrt[41]

So, the radius of the circle is sqrt[41].