The equation of a circle:
\((x-h)^2+(y-k)^2=r^2\)
(h,k) - the coordinates of the center
r - the radius
\(\hbox{the center: } A(-3,2) \\
h=-3 \\ k=2 \\ \\
\hbox{the equation:} \\
(x+3)^2+(y-2)^2=r^2 \\ \\
\hbox{the circle passes through B(1,3)} \\
x=1 \\ y=3 \\ \Downarrow \\
(1+3)^2+(3-2)^2=r^2 \\
4^2+1^2=r^2 \\
16+1=r^2 \\
17=r^2 \\ \\
\hbox{the equation is:} \\
(x+3)^2+(y-2)^2=17\)
Plug the coordinates of the points into the equation and check:
\(C(-1,-2) \\
(-1+3)^2+(-2-2)^2=17 \\
2^2+(-4)^2=17 \\
4+16=17 \\
20=17 \\
not \ true \\ \\
D(-6,3) \\
(-6+3)^2+(3-2)^2=17 \\
(-3)^2+1^2=17 \\
9+1=17 \\
10=17 \\
not \ true\)
\(E(-3,-3) \\
(-3+3)^2+(-3-2)^2=17 \\
0^2+(-5)^2=17 \\
25=17 \\
not \ true \\ \\
F(-2,6) \\
(-2+3)^2+(6-2)^2=17 \\
1^2+4^2=17 \\
1+16=17 \\
17=17 \\
true\)
The answer is F(-2,6).
