Given a circle where the center is (-2,2) and another point on the circle is (-10,-4)

Given a circle with its center at \((-2, 2)\) and a point on the circle at \((-10, -4)\), we can calculate several properties of the circle, such as its radius and equation. Here's the step-by-step process:

### 1. Finding the radius of the circle

The radius \(r\) is the distance between the center \((-2, 2)\) and the point on the circle \((-10, -4)\). The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula:

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

Substituting the coordinates:

\[
r = \sqrt{(-10 - (-2))^2 + (-4 - 2)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

So, the radius \(r = 10\).

### 2. Equation of the circle

The general equation of a circle with center \((h, k)\) and radius \(r\) is:

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

Substitute \(h = -2\), \(k = 2\), and \(r = 10\):

\[
(x - (-2))^2 + (y - 2)^2 = 10^2
\]

Simplifying:

\[
(x + 2)^2 + (y - 2)^2 = 100
\]

Thus, the equation of the circle is:

\[
(x + 2)^2 + (y - 2)^2 = 100
\]

### Summary:
- **Center**: \((-2, 2)\)
- **Radius**: 10
- **Equation of the circle**: \((x + 2)^2 + (y - 2)^2 = 100\)

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do you find the distance between two points in 3D space?
2. How can you find the circumference of this circle?
3. How would you calculate the area of this circle?
4. What is the equation of a tangent line at a given point on this circle?
5. How do you find the length of a chord of a circle?

**Tip:** The general form of a circle's equation is useful for graphing and analyzing relationships between points on the circle.

Learn how to find the radius and the equation of a circle given its center at (-2, 2) and a point on the circle at (-10, -4). This solution includes key geometric concepts such as the distance formula and the standard equation of a circle. Suitable for students in Grades 9-10.

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