Acircle has a diameter with endpoints (-7, -1) and (-5, -9). what is the equation of the circle?

We know that

the general equation of a circle is
(x-h)²+(y-k)²=r²
where
the center is the point (h,k)
the radius is r

the circle has a diameter with endpoints (-7, -1) and (-5, -9)
Let
A (-7, -1)
B (-5, -9)
the distance between A and B is the diameter
d=√[(-8)²+(2)²]----------> d=√68
diameter=√68--------------> r=√68/2 units

the center of a circle is the midpoint between A and B
Xm=(-7-5)/2--------> -6
Ym=(-9-1)/2---------> -5

the center is the point (-6,-5)

(x-h)²+(y-k)²=r²----------> (x+6)²+(y+5)²=(√68/2)²
(x+6)²+(y+5)²=68/4--------> (x+6)²+(y+5)²=17

the answer is 
(x+6)²+(y+5)²=17

see the attached figure

It's like discovering the answer to a puzzle. the purpose in solving an equation is to find the value or values of the variable that makes it a true statement. any value of the variable that makes the equation true is called a solution to the equation.