The endpoints of a diameter lie on opposite sides of the circle, with the center exactly halfway between them. How can you find the midpoint between $(2,-3)$ and $(8,9)$?

Once you know the center, use the distance formula between the center and one endpoint of the diameter to find the radius. Remember you only need $r^2$ for the equation.

In Desmos, enter the points (2,-3) and (8,9) on separate lines. This lets you see the segment that is the diameter of the circle.

On a new line, type (2+8)/2 and on the next line type (-3+9)/2. Desmos will show the $x$- and $y$-coordinates of the midpoint; together they give the center of the circle.

On a new line, type ((8-2)^2 + (9-(-3))^2)/4. This calculates one quarter of the squared distance between the endpoints, which equals the radius squared.

Use the center you found in step 2 as $(h,k)$ and the value from step 3 as $r^2$. Look at each answer choice’s form $(x-h)^2+(y-k)^2=r^2$ and select the one whose center and $r^2$ match these values.

Any circle with center at $(h,k)$ and radius $r$ has equation

So for this problem, we need to find the center $(h,k)$ and the radius $r$ of the circle whose diameter has endpoints $(2,-3)$ and $(8,9)$.

The center of the circle is the midpoint of the diameter segment connecting $(2,-3)$ and $(8,9)$.

Use the midpoint formula: the midpoint of $(x_1, y_1)$ and $(x_2, y_2)$ is

Here:

• $x_1 = 2$, $x_2 = 8$ so the $x$-coordinate of the center is $\dfrac{2+8}{2} = \dfrac{10}{2} = 5$.
• $y_1 = -3$, $y_2 = 9$ so the $y$-coordinate of the center is $\dfrac{-3+9}{2} = \dfrac{6}{2} = 3$.

So the center of the circle is $(5,3)$.

The radius is the distance from the center to either endpoint of the diameter. We only need $r^2$, so we can use the distance formula without taking the square root.

Using the center $(5,3)$ and endpoint $(2,-3)$:

Compute each part:

• $5 - 2 = 3$, so $(5 - 2)^2 = 3^2 = 9$.
• $3 - (-3) = 3 + 3 = 6$, so $(3 - (-3))^2 = 6^2 = 36$.

So

Now plug $h = 5$, $k = 3$, and $r^2 = 45$ into the standard form

to get

This matches answer choice C, $(x-5)^2+(y-3)^2=45$.