Question

Which point lies on a circle that is centered at A(-3, 2) and passes through B(1, 3)?

Answer 1

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).