https://file.aiquickdraw.com/mathbot/file/1725397136134freyx0zp.jpg

The problem asks for the equation of a circle centered at \((-7, -2)\) and passing through the point \((-7, 4)\).

### Step 1: Determine the Radius
The radius \(r\) of the circle can be found using the distance formula between the center \((-7, -2)\) and the point \((-7, 4)\) on the circle:
\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the given points:
\[
r = \sqrt{(-7 - (-7))^2 + (4 - (-2))^2} = \sqrt{0^2 + (6)^2} = \sqrt{36} = 6
\]

### Step 2: Write the Equation of the Circle
The general form for 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. Plugging in the center \((-7, -2)\) and the radius \(r = 6\):
\[
(x + 7)^2 + (y + 2)^2 = 36
\]

So, the equation of the circle is:
\[
(x + 7)^2 + (y + 2)^2 = 36
\]

Would you like further details or explanations on any part of this solution?

### Related Questions:
1. How do you find the distance between two points in a plane?
2. How do you derive the equation of a circle from its general form?
3. What is the significance of the radius in the equation of a circle?
4. How can you verify if a point lies on a given circle?
5. What are the standard and general forms of the equation of a circle?

### Tip:
Always check your final equation by plugging in the given points to ensure they satisfy the equation. This helps confirm that your derived equation is correct.

Find the equation of a circle centered at (-7, -2), passing through the point (-7, 4).

This problem involves finding the equation of a circle centered at (-7, -2) and passing through the point (-7, 4). The solution includes applying the distance formula to find the radius and then using the standard form equation of a circle. Suitable for students in Grades 9-10.

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