Circles SAT question

Choice D is correct. A circle in the xy-plane can be represented by an equation of the form ${(x-h)}^2+{(y-k)}^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the length of a radius of the circle. It's given that the circle has its center at $(-4,5)$. Therefore, $h=-4$ and $k=5$. Substituting $-4$ for $h$ and $5$ for $k$ in the equation ${(x-h)}^2+{(y-k)}^2=r^2$ yields ${(x-(-4))}^2+{(y-5)}^2=r^2$, or ${(x+4)}^2+{(y-5)}^2=r^2$. It's also given that the point $(-8,8)$ lies on the circle. Substituting $-8$ for $x$ and $8$ for $y$ in the equation ${(x+4)}^2+{(y-5)}^2=r^2$ yields ${(-8+4)}^2+{(8-5)}^2=r^2$, or ${(-4)}^2+{(3)}^2=r^2$, which is equivalent to $16+9=r^2$, or $25=r^2$. Substituting $25$ for $r^2$ in the equation ${(x+4)}^2+{(y-5)}^2=r^2$ yields ${(x+4)}^2+{(y-5)}^2=25$. Thus, the equation ${(x+4)}^2+{(y-5)}^2=25$ represents the circle.

Choice A is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.

Choice B is incorrect. The point $(-8,8)$ doesn't lie on the circle represented by this equation.

Choice C is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.