Determine the equation of the circle with center left parenthesis, minus, 6, comma, 2, right parenthesis(−6,2) containing the point left parenthesis, minus, 9, comma, minus, 2, right parenthesis(−9,−2).

To determine the equation of the circle, we start with the center \((h, k) = (-6, 2)\) and a point on the circle \((x, y) = (-9, -2)\).

1. **Calculate the radius** using the distance formula:
   \[
   r = \sqrt{(x - h)^2 + (y - k)^2}
   \]
   Substituting the values:
   \[
   r = \sqrt{(-9 - (-6))^2 + (-2 - 2)^2} = \sqrt{(-9 + 6)^2 + (-2 - 2)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
   \]

2. **Use the standard form of the circle's equation**:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   Substituting \(h = -6\), \(k = 2\), and \(r = 5\):
   \[
   (x + 6)^2 + (y - 2)^2 = 5^2
   \]
   Simplifying:
   \[
   (x + 6)^2 + (y - 2)^2 = 25
   \]

Thus, the equation of the circle is:
\[
(x + 6)^2 + (y - 2)^2 = 25
\]

### Questions or Details
Do you want further details on any steps, or do you have any questions? Here are some related questions for you:

1. How can you derive the center and radius from the standard form of a circle's equation?
2. What does the graph of this circle look like?
3. How would you find the equation of a circle given two points on its circumference?
4. How can you determine if a point lies inside, outside, or on the circle?
5. What are the implications of the radius being larger or smaller in the context of the circle's size? 

**Tip**: Always visualize the problem by sketching the circle and marking the center and points to better understand its properties.

Determine the equation of the circle with center (-6, 2) containing the point (-9, -2).

Learn how to determine the equation of a circle with center (-6, 2) that passes through the point (-9, -2). This guide includes step-by-step calculations using the distance formula and the standard form of a circle's equation, suitable for Grades 8-10.

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