The two given points are endpoints of a diameter. How can you use these two points to find the center of the circle?

Once you know the distance between the two given points, what does that distance represent, and how do you turn it into the radius (and then $r^2$) for the circle equation?

Enter the two points as (2,-1) and (-6,5) in Desmos so you can see the segment that is supposed to be a diameter of the circle.

On separate lines, type each option exactly as written, for example y^2+4y+x^2-4x=10 after expanding, or more simply use implicit form like (x-2)^2+(y+2)^2=10, so that each choice appears as a circle on the graph.

Look for the circle on which both points (2,-1) and (-6,5) lie, and such that the line segment between these points passes through the circle’s center (looks like a straight line cutting the circle exactly in half). The equation of that circle matches the correct answer choice.

A circle with center $(h,k)$ and radius $r$ in the coordinate plane has equation

So we need to find the center $(h,k)$ and the radius $r$ from the endpoints of the diameter $(2,-1)$ and $(-6,5)$.

The center of a circle is the midpoint of any diameter. Use the midpoint formula for points $(x_1,y_1)$ and $(x_2,y_2)$:

Here, $(x_1,y_1) = (2,-1)$ and $(x_2,y_2) = (-6,5)$, so

So $h=-2$ and $k=2$.

The distance between the two endpoints of the diameter is the length of the diameter. Use the distance formula:

The radius is half the diameter, so

We have center $(-2,2)$ and $r^2 = 25$. Substitute into $(x-h)^2 + (y-k)^2 = r^2$:

which simplifies to

This matches answer choice B, so the correct equation is $(x+2)^2+(y-2)^2=25$.