The problem asks to find the equation of a circle given its graph. We can identify two points on the circle: $(3, 6)$ and $(3, -4)$.

1. Problem Description

The problem asks to find the equation of a circle given its graph. We can identify two points on the circle: $(3, 6)$ and $(3, -4)$.

2. Solution Steps

First, we need to find the center of the circle. Since we have two points on the circle with the same x-coordinate, we know that the center must lie on the vertical line $x = 3$. The y-coordinate of the center will be the midpoint of the y-coordinates of the two given points.

The y-coordinate of the center is:

$y_c = \frac{6 + (-4)}{2} = \frac{2}{2} = 1$

So, the center of the circle is $(3, 1)$.

Next, we need to find the radius of the circle. The radius is the distance from the center to any point on the circle. Let's use the point $(3, 6)$. The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

In this case, $(x_1, y_1) = (3, 1)$ and $(x_2, y_2) = (3, 6)$, so the radius is:

$r = \sqrt{(3 - 3)^2 + (6 - 1)^2} = \sqrt{0^2 + 5^2} = \sqrt{25} = 5$

Thus, the radius of the circle is

5.

The general equation of a circle with center $(h, k)$ and radius $r$ is:

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

In our case, $(h, k) = (3, 1)$ and $r = 5$, so the equation of the circle is:

$(x - 3)^2 + (y - 1)^2 = 5^2$

$(x - 3)^2 + (y - 1)^2 = 25$

3. Final Answer

$(x - 3)^2 + (y - 1)^2 = 25$