Use the midpoint formula on $A(-6,5)$ and $B(2,-3)$ to find the center of the circle.

After finding the distance between $A$ and $B$, remember that this is the diameter. How do you get the radius (and $r^2$) from the diameter?

Once you know the center $(h,k)$ and $r^2$, rewrite $(x-h)^2 + (y-k)^2 = r^2$ and compare it to the answer choices.

Type (-6,5) and (2,-3) into Desmos on separate lines to plot points $A$ and $B$.

On a new line, type ((-6+2)/2, (5+(-3))/2) to create a point for the midpoint of $AB$. This point is the center of the circle; note its coordinates from the graph or from the expression list.

Type d = distance[(-6,5),(2,-3)] to have Desmos compute the diameter length. Then type r = d/2 for the radius and r^2 on a new line to see the value of $r^2$.

Using the center coordinates $(h,k)$ you found and the value of r^2, enter (x - h)^2 + (y - k)^2 = r^2 into Desmos. Compare the resulting equation (signs on $h$ and $k$ and the value of $r^2$) with the answer choices to see which one matches.

The standard (center-radius) form of a circle in the $xy$-plane is

where $(h,k)$ is the center and $r$ is the radius. So we need to find the center and radius of the circle whose diameter has endpoints $A(-6,5)$ and $B(2,-3)$.

The center of a circle is the midpoint of any diameter.

Use the midpoint formula for points $A(x_1,y_1)$ and $B(x_2,y_2)$:

For $A(-6,5)$ and $B(2,-3)$:

So the center of the circle is $(-2,1)$.

The diameter is the distance between $A$ and $B$, and the radius is half of that distance.

First find the distance $AB$ using the distance formula:

For $A(-6,5)$ and $B(2,-3)$:

So the diameter length is $\sqrt{128}$. The radius is half of that:

We will use $r^2$ in the circle equation, so square the radius:

Now plug the center $(-2,1)$ and $r^2 = 32$ into the standard form $(x - h)^2 + (y - k)^2 = r^2$.

• Since $h = -2$, $x - h = x - (-2) = x + 2$.
• Since $k = 1$, $y - k = y - 1$.

So the equation of the circle is