A circle has a diameter with endpoints (8, 2) and (2, 6). What is the equation of the circle?

To find the equation of the circle, we first need to determine the center and the radius of the circle.

1. **Finding the Center**: The center of the circle is the midpoint of the diameter. The midpoint formula is:

Midpoint, M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Using the endpoints (8, 2) and (2, 6):

M = ((8 + 2) / 2, (2 + 6) / 2) = (10 / 2, 8 / 2) = (5, 4)

So, the center of the circle is (5, 4).

2. **Finding the Radius**: The radius is half the length of the diameter. We can calculate the length of the diameter using the distance formula:

Distance, d = √((x₂ – x₁)² + (y₂ – y₁)²)

Applying this to our endpoints:

d = √((2 – 8)² + (6 – 2)²) = √((-6)² + (4)²) = √(36 + 16) = √52 = 2√13

The radius, r, is half of this distance:

r = d / 2 = (2√13) / 2 = √13

3. **Writing the Equation**: The standard equation of a circle with center (h, k) and radius r is:

(x – h)² + (y – k)² = r²

Plugging in our values:

(x – 5)² + (y – 4)² = (√13)²

(x – 5)² + (y – 4)² = 13

Thus, the equation of the circle is:

(x – 5)² + (y – 4)² = 13