The equation of circle having a diameter with endpoints (-2, 1) and (6, 7) is
(x - 4)² + (y - 3)² = 25
(x - 2)² + (y - 4)² = 25
(x - 2)² + (y - 4)² = 100

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

The 'x' coordinate of the midpoint is  (1/2) (6 - 2) = 2
The 'y' coordinate of the midpoint is  (1/2) (7 + 1) = 4
The center of the circle is the point  (2, 4) .

The length of the diameter of the circle is the distance
between the endpoints of the diameter.

The distance between (-2, 1) and (6, 7) is

                              √ (8² + 6²)

                           = √ (64 + 36)

                           = √ 100  =  10 .

The radius of the circle is      1/2 of 10  =  5 .

Now, the equation of any circle is

        (x - [x of the center])² + (y - [y of the center])² = (radius)² .

and we have all the numbers now.

                    (x - 2)² + (y - 4)²  =  (5)² .

I'm very happy to see that this equation is actually
one of the choices !     I'll bet you are too.

Step-by-step explanation:

In the picture you can see that has endpoints of (-2, 1) and (6, 7)