If the coordinates of the endpoints of a diameter of the circle are known, the equation of a circle can be found. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius, which is the distance from the center to either endpoint of the diameter. Finally use the center-radius form to find the equation Find the center-radius form for the circle having the endpoints (1,4) and (7,6) of a diameter

(x - 4)² + (y - 5)² = 10

Step-by-step explanation:

First, we find the coordinates of the center of the circle by the formula ((x₁ + x₂)/2, (y₁ + y₂)/2). The point (x₁,y₁) = (1,4) and (x₂,y₂) = (7,6). So, the coordinates of the center of the circle are the coordinates of the midpoint of the diameter. So, ((1 + 7)/2,(4 + 6)/2) = (8/2, 10/2) = (4,5). We then find the radius which is the distance from the center of the circle to any of the end points of the diameter. So, r² = (x₂ - x₁)² + (y₂ - y₁)². Let (x₁, y₁) = (4,5) and (x₂, y₂) = (7,6). So, r² = (7 -4)² + (6 - 5)² = 3² + 1² = 9 + 1 = 10.

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

With center (h,k) = (4,5) and r² = 10, we have

(x - 4)² + (y - 5)² = 10