Find an equation of the circle that satisfies the given conditions.

endpoints of a diameter at 

(−1, 1)

 and 

(7, 9)

To find the equation of a circle with a diameter that has endpoints at \((-1, 1)\) and \((7, 9)\), we need to determine two key properties of the circle:

1. **The center of the circle**: This is the midpoint of the diameter.
2. **The radius of the circle**: This is half the length of the diameter.

### Step 1: Find the center of the circle
The center \(C(x, y)\) of the circle is the midpoint of the diameter's endpoints. The midpoint formula is:

\[
C(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

For the given points \((-1, 1)\) and \((7, 9)\):

\[
C(x, y) = \left( \frac{-1 + 7}{2}, \frac{1 + 9}{2} \right) = \left( \frac{6}{2}, \frac{10}{2} \right) = (3, 5)
\]

So, the center of the circle is at \((3, 5)\).

### Step 2: Find the radius of the circle
The radius \(r\) is half the length of the diameter. First, we find the length of the diameter using the distance formula:

\[
\text{Diameter} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates:

\[
\text{Diameter} = \sqrt{(7 - (-1))^2 + (9 - 1)^2} = \sqrt{(7 + 1)^2 + (8)^2} = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}
\]

The radius \(r\) is half the diameter:

\[
r = \frac{8\sqrt{2}}{2} = 4\sqrt{2}
\]

### Step 3: Write the equation of the circle
The standard form of the equation of a circle is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Where \((h, k)\) is the center and \(r\) is the radius.

Substituting \(h = 3\), \(k = 5\), and \(r = 4\sqrt{2}\):

\[
(x - 3)^2 + (y - 5)^2 = (4\sqrt{2})^2 = 32
\]

So, the equation of the circle is:

\[
(x - 3)^2 + (y - 5)^2 = 32
\]

Would you like to dive deeper into any part of the solution or have further questions? Here are some related questions you could explore:

1. How would you find the equation of a circle given the radius and center?
2. What is the significance of the circle's general equation \((x-h)^2 + (y-k)^2 = r^2\)?
3. How do you derive the equation of a circle from its general form \(Ax^2 + Ay^2 + Bx + Cy + D = 0\)?
4. How do changes to the radius and center affect the graph of the circle?
5. Can you calculate the area of the circle using the radius?

**Tip:** Always remember to square the radius when you write the equation of the circle. It’s a common mistake to forget this step!

Learn how to find the equation of a circle with diameter endpoints at (-1, 1) and (7, 9). This detailed solution covers the midpoint formula, distance formula, and the step-by-step process to derive the circle's equation. Suitable for high school students studying coordinate geometry.

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