The equation of a circle with center $(h,k)$ and radius $r$ is $(x-h{)}^2+(y-k{)}^2=r^2$. The fountain’s center is $(4,-2)$, so the equation begins as $(x-4{)}^2+(y+2{)}^2=r^2$. The diameter is 18 feet, so the radius is half of that: $r=9$. Then $r^2=9^2=81$. Substituting gives $(x-4{)}^2+(y+2{)}^2=81$, which matches choice C.

Start with $x^2+y^2-8x+6y-11=0$. Move the constant to the other side: $x^2-8x+y^2+6y=11$. Complete the square for each variable: $x^2-8x=(x-4{)}^2-16$ and $y^2+6y=(y+3{)}^2-9$. Substitute these expressions: $(x-4{)}^2-16+(y+3{)}^2-9=11$. Combine constants: $(x-4{)}^2+(y+3{)}^2-25=11$. Add 25 to both sides to get $(x-4{)}^2+(y+3{)}^2=36$. This matches choice A.

The standard form of a circle with center $(h,k)$ and radius $r$ is $(x-h{)}^2+(y-k{)}^2=r^2$. Here, the center is $(3,-2)$, so $h=3$ and $k=-2$. Substituting gives $(x-3{)}^2+(y-(-2){)}^2=4^2$, which simplifies to $(x-3{)}^2+(y+2{)}^2=16$. Therefore, choice A is correct.

The standard form of a circle is $(x-h{)}^2+(y-k{)}^2=r^2$, where $(h,k)$ is the center. Since the center is $(2,-1)$, the equation must be $(x-2{)}^2+(y+1{)}^2=r^2$. To find $r^2$, use the point $(5,3)$ on the circle:

So the equation is $(x-2{)}^2+(y+1{)}^2=25$.

The equation of a circle with center $(h,k)$ is $(x-h{)}^2+(y-k{)}^2=r^2$. Since the center is $(1,1)$, the equation must have the form $(x-1{)}^2+(y-1{)}^2=r^2$. Use any point from the table, such as $P=(4,1)$, to find $r^2$:

So $r^2=9$, and the equation is $(x-1{)}^2+(y-1{)}^2=9$.