Find the equation (formula) of a circle with radius r and center C(h,k) and if the Center of a circle is at (7,-2) and a point on the circle is (-3,5) find the formula of the circle.

### Equation of a Circle

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

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

### Given Problem

You are given:
- The center of the circle: \( C(7, -2) \)
- A point on the circle: \( (-3, 5) \)

First, let's calculate the radius \( r \) using the distance formula between the center \( C(7, -2) \) and the given point \( (-3, 5) \):

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

Substituting the values:

\[
r = \sqrt{(-3 - 7)^2 + (5 + 2)^2}
\]
\[
r = \sqrt{(-10)^2 + (7)^2}
\]
\[
r = \sqrt{100 + 49} = \sqrt{149}
\]

Now, we have the radius \( r = \sqrt{149} \).

### Formula of the Circle

Substituting the center \( C(7, -2) \) and radius \( r = \sqrt{149} \) into the general equation:

\[
(x - 7)^2 + (y + 2)^2 = (\sqrt{149})^2
\]

Simplifying the equation:

\[
(x - 7)^2 + (y + 2)^2 = 149
\]

### Final Answer

The equation of the circle is:

\[
(x - 7)^2 + (y + 2)^2 = 149
\]

Would you like further details or have any questions?

Here are some related questions:
1. How can you find the radius if only the equation of the circle is given?
2. What is the meaning of each term in the circle's equation?
3. How would the equation change if the center was at the origin?
4. How do you determine if a point lies inside, on, or outside the circle?
5. How can you find the diameter of the circle using its equation?

**Tip:** The distance formula is essential for finding the radius of a circle when you have the center and a point on the circle.

Learn how to find the equation of a circle with center at (7, -2) and a point (-3, 5) on the circle. Includes step-by-step calculations using the distance formula and equation of a circle formula. Suitable for high school students studying geometry.

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