How do you find the equation of the circle with a diameter defined by the endpoints (7, 3) and (7, 5)?

To find the equation of a circle given the endpoints of its diameter, you first need to determine the center and the radius of the circle.

1. **Find the center of the circle**: The center is the midpoint of the diameter. We can calculate the midpoint (M) using the following formula:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Here, our endpoints are (7, 3) and (7, 5). Substituting these values into the formula gives:

M = ((7 + 7) / 2, (3 + 5) / 2) = (7, 4).

So, the center of the circle is (7, 4).

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

d = √((x₂ – x₁)² + (y₂ – y₁)²).

Substituting the endpoints, we get:

d = √((7 – 7)² + (5 – 3)²) = √(0 + 4) = √4 = 2.

Since the radius is half the diameter, we have:

Radius (r) = d / 2 = 2 / 2 = 1.

3. **Writing the equation of the circle**: The standard form of the equation of a circle is:

(x – h)² + (y – k)² = r²,

where (h, k) is the center and r is the radius. Plugging in our values, we have:

(x – 7)² + (y – 4)² = 1².

Therefore, the equation of the circle is:

(x – 7)² + (y – 4)² = 1.