sat-dog

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

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

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

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