To find the equation of the circle, we use the standard form of the equation of a circle:

$(x - h)^2 + (y - k)^2 = r^2$

Where:

• $(h, k)$ is the center of the circle,
• $r$ is the radius.

Given:

• The center of the circle is $(-3, 2)$, so $h = -3$ and $k = 2$.
• A point on the circle is $(6, 4)$, which we can use to calculate the radius $r$.

Step 1: Calculate the radius

The radius is the distance between the center $(-3, 2)$ and the point $(6, 4)$, and we can find this distance using the distance formula:

$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substitute the coordinates of the center and the point on the circle:

$r = \sqrt{(6 - (-3))^2 + (4 - 2)^2}$ $r = \sqrt{(6 + 3)^2 + (4 - 2)^2}$ $r = \sqrt{9^2 + 2^2}$ $r = \sqrt{81 + 4} = \sqrt{85}$

So, $r^2 = 85$.

Step 2: Write the equation of the circle

Now that we have the center $(-3, 2)$ and $r^2 = 85$, we can substitute into the equation:

$(x - (-3))^2 + (y - 2)^2 = 85$ $(x + 3)^2 + (y - 2)^2 = 85$

This matches option C.

Answer:

The correct equation is:

C. $(x + 3)^2 + (y - 2)^2 = 85$

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Would you like to explore further details or ask about another topic?

Here are 5 related questions:

1. How do you calculate the distance between two points in a 2D plane?
2. What is the general form of a circle's equation in 2D geometry?
3. How does the center-radius form of a circle help in graphing the circle?
4. How can you determine if a point lies inside or outside a circle using the equation?
5. Can you derive the equation of a circle given just two points on its circumference?

Tip: Always double-check your distance calculations, as small errors can throw off the radius and thus the entire equation.