find the equation of a circle. the endpoints of the diameter of a circle are (3,-2) and (7,-6).

centre equals the midpoint = (5,-4)

equation:
(x-5)^2 + (y+4)^2 = r^2
plug in (3,-2)
4 + 4 = r^2 = 8

(x-5)^2 + (y+4)^2 = 8

Those are endpoints of A diameter which is neither horizontal nor vertical. There are other possible diameter endpoint pairs for the same circle.

The midpoint of the given diameter is at (5,-4). You get that by averaging the two x's and the two y's, separately.

The length of the diameter is
D = sqrt(4^2 + 4^2) = sqrt32 = 4sqrt2
The radius is R = 2sqrt2

The equation of the circle is therefore
(x-5)^2 + (y+4)^2 = R^2 = 8
or
(x-5)^2/8 + (y+4)^2/8 = 1

Check: when x = 3 and y = -2
4/8 + 4/8 = 1

I got everything the same except the radius. How did you get 8?

The radius is not 8. It is the square root of 8.

To find the equation of a circle, you need the coordinates of its center and the length of its radius. In this case, you are given the endpoints of the diameter of the circle, which can help us find the center of the circle.

1. Find the midpoint of the diameter:
Using the midpoint formula, we can find the coordinates of the center of the circle by finding the average of the x-coordinates and the average of the y-coordinates of the two endpoints.
The midpoint coordinates can be calculated as follows:

Midpoint(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the diameter.

In this case, the endpoints of the diameter are (3, -2) and (7, -6):
Midpoint(x, y) = ((3 + 7)/2, (-2 - 6)/2)
Midpoint(x, y) = (10/2, -8/2)
Midpoint(x, y) = (5, -4)

Hence, the center of the circle is (5, -4).

2. Find the radius of the circle:
The radius of the circle is half the length of the diameter. To find the length of the diameter, we can use the distance formula.

Distance between two points (x1, y1) and (x2, y2) is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, the two endpoints of the diameter are (3, -2) and (7, -6):
Distance = √((7 - 3)^2 + (-6 - (-2))^2)
Distance = √((4)^2 + (-4)^2)
Distance = √(16 + 16)
Distance = √32

Since the radius is half the length of the diameter:
Radius = √32 / 2
Radius = √8
Radius = 2√2

Hence, the radius of the circle is 2√2.

3. Write the equation of the circle:
The equation of a circle with its center at (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2

In this case, the center of the circle is (5, -4) and the radius is 2√2:
(x - 5)^2 + (y - (-4))^2 = (2√2)^2
(x - 5)^2 + (y + 4)^2 = 8

Hence, the equation of the circle is (x - 5)^2 + (y + 4)^2 = 8.