1. which is the equation of a circle with center (–2, 3) and radius r = 5?
(x + 2)² + (y – 3)² = 10
(x + 2)² + (y – 3)² = 25
(x – 2)² + (y + 3)² = 10
(x – 2)² + (y + 3)² = 25

2. a circle with center (–1, 2) passes through point (2, –2). which is true?
the radius is sqrt 5.
the diameter is 10.
the equation is (x + 1)² + (y – 2)² = 10.
the circumference is 25.

4. which is the equation of a circle with diameter with a(5, 4) and b(–1, –4)?
(x – 5)² + (y – 4)² = 10
(x + 5)² + (y + 4)² = 100
(x – 2)² + y² = 25
(x + 2)² + y² = 5

1) the main formula of circle equation is (x - a)² + (y – b)²= R², (a, b) is the center and R is the radius. So we replace easily (a, b) by (–2, 3), and R=5 for finding the answer, it is (x +2 )² + (y – 3)²= 5²= 25
2) if a circle (x - a)² + (y – b)²= R² , ((a, b) is the center) passes with A(u, v), so we can write the equation as (u- a)² + (v – b)²= R², so we have 
(2 + 1)² + (-2 -2)² = R², R²=9+16=25, which implies R=5, and the diameter is given by D=2*R=10, so the answer is The diameter is 10.
3) firstly, D=AB, let's calculate the length of AB,
AB= sqrt[(5+1)^2+(4+4)^2]=sqrt[36+64]=sqrt[100], so D=AB=10,           but we know that D=R*2, so R=10/2=5, the equation of the circle must be as follow (x - a)² + (y – b)²= R², after changing the value of R, the equation is 
(x – 2)² + y² = 25

The y-intercept is at (0,32)