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We are given two endpoints of a diameter of a circle: \((-2, -1)\) and \((-2, 7)\).

### a.) Find the center of the circle.
The center of a circle that has its diameter defined by two points is the midpoint of those two points. The midpoint \( M \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using:

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

Using the endpoints \((-2, -1)\) and \((-2, 7)\):
\[
M = \left( \frac{-2 + (-2)}{2}, \frac{-1 + 7}{2} \right) = (-2, 3)
\]

So, the center of the circle is \((-2, 3)\).

### b.) Determine the radius of the circle.
The radius is half the length of the diameter. We can find the length of the diameter by calculating the distance between the two points \((-2, -1)\) and \((-2, 7)\) using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the values:
\[
d = \sqrt{(-2 - (-2))^2 + (7 - (-1))^2} = \sqrt{(0)^2 + (8)^2} = 8
\]

Since the diameter is 8 units, the radius is:
\[
\text{Radius} = \frac{8}{2} = 4
\]

### c.) Write the equation of the circle.
The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.

From parts (a) and (b), the center is \((-2, 3)\) and the radius is \(4\). Plugging these into the equation:

\[
(x - (-2))^2 + (y - 3)^2 = 4^2
\]
which simplifies to:
\[
(x + 2)^2 + (y - 3)^2 = 16
\]

This is the equation of the circle.

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Here are five related questions to expand on this:
1. How do you find the center if only one point and the radius are given?
2. Can a circle have a diameter along a non-vertical or non-horizontal line?
3. How do you find the equation of a circle given its center and circumference?
4. What is the relationship between the diameter and circumference of a circle?
5. How can you find the area of a circle from its equation?

**Tip:** For problems involving the distance between two points, always recall the distance formula.

Sresha is trying to find the equation of a circle. She knows that two endpoints of a diameter are (-2, -1) and (-2, 7). a) Find the center of the circle. b) Determine the radius of the circle. c) Write the equation of the circle.

Learn how to find the equation of a circle when given the endpoints of its diameter. This problem involves calculating the midpoint to find the center, using the distance formula to determine the radius, and forming the equation of the circle. Suitable for students in Grades 9-10.

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