Question

1. Which is the equation of a circle with center (–2, 3) and radius r = 5?

Answer 1

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