The standard form of a circle's equation is $(x-h{)}^2+(y-k{)}^2=r^2$, where $(h,k)$ is the center. Since the center is $(4,-1)$, the equation must begin as $(x-4{)}^2+(y+1{)}^2=r^2$. Next, find $r^2$ using the point $(10,7)$ on the circle:

Therefore, the equation is $(x-4{)}^2+(y+1{)}^2=100$.

Because the circle passes through $(6,2)$ and $(2,6)$, its center must be equidistant from those two points. The center is also on the line $y=x$. The midpoint of $(6,2)$ and $(2,6)$ is $\left(\frac{6+2}{2},\frac{2+6}{2}\right)=(4,4)$, and this point lies on $y=x$, so the center is $(4,4)$. The radius is the distance from $(4,4)$ to either given point: $r=\sqrt{(6-4{)}^2+(2-4{)}^2}=\sqrt{4+4}=\sqrt{8}$. Therefore, the equation is $(x-4{)}^2+(y-4{)}^2=8$.

The points come in opposite pairs: $P(1,3)$ and $Q(5,3)$ lie on the same horizontal line, and $R(3,5)$ and $S(3,1)$ lie on the same vertical line. The center of the circle is the midpoint of either pair. Using $P$ and $Q$, the midpoint is $\left(\frac{1+5}{2},\frac{3+3}{2}\right)=(3,3)$. So the center is $(3,3)$. The radius is the distance from the center to any listed point, such as from $(3,3)$ to $(1,3)$, which is $2$. A circle with center $(h,k)$ and radius $r$ has equation $(x-h{)}^2+(y-k{)}^2=r^2$. Therefore, the equation is $(x-3{)}^2+(y-3{)}^2=2^2=4$.

The circle is centered at $(2,-1)$ and passes through $(8,7)$, so its radius is the distance from the center to that point:

The bench is directly east of the fountain, so it is $10$ units to the right of the center: $(12,-1)$. The sculpture is directly north of the fountain, so it is $10$ units above the center: $(2,9)$.
The point directly northeast of the fountain that forms the rectangle with these two points is $(12,9)$. Thus, the rectangle has width $12-2=10$ and height $9-(-1)=10$. Its area is

So the correct answer is $100$.

Because the circular path passes through $(2,6)$ and $(8,6)$, the center must lie on the perpendicular bisector of the segment joining those points. The midpoint of the segment is

Since the segment is horizontal, its perpendicular bisector is the vertical line $x=5$. The center is also given to lie on $y=2x-6$, so substitute $x=5$ into that line:

Therefore, the center is $(5,4)$. The radius is the distance from $(5,4)$ to either given point, such as $(2,6)$:

So the equation is

which is choice A.