A diameter of a circle has endpoints P(-10,-2) and Q(4,6)
Find the center of the circle
Find the radius. If your answer is not an integer, express it in radical form
Write an equation for the circle.

Solution:

Given endpoints of a circle (-10, -2) and (4, 6)

1. We know that the center of the circle is the midpoint of the diameter line formed by two endpoints

Centre C = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

Here, x₁ = -10; y₁ = -2; x₂ = 4; y₂ = 6

(h, k) = (-10 + 4/2, -2 + 6/2)

(h, k) =(-6/2, 4/2)

(h, k) = (-3, 2)

Therefore, the centre is (-3, 2)

2. The radius can be calculated as the distance between the centre and the endpoint of the diameter

We have radius r = √{ (x₂ - x₁)² + (y₂ - y₁)²}

Here, x₁ = -3; y₁ = 2; x₂ = 4; y₂= 6

r = √{(4 - (-3))² + (6 - 2)²}

r = √{(4 + 3)² + (6 - 2)²}

r = √{(7)² + (4)²}

r = √{49 + 16}

r = √65

Radius of circle is √65 units.

3. The equation of a circle with centre (h, k) and radius r is given as (x - h)² + (y - k)² = r²

Here, (h, k) = (-3, 2) and radius r = √65

(x - (-3))² + (y - 2)² = (√65)²

(x + 3)² + (y - 2)² = 65

The equation of the circle is (x + 3)² + (y - 2)² = 65