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: $(x + 7)^2 + (y + 2)^2 = 36$

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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.