From $(x-3)^2 + (y+4)^2 = 81$, determine the circle’s center coordinates and its radius before any translation.

If a point moves 2 units to the right and 5 units down, how do its $x$- and $y$-coordinates change? Apply that to the original center.

Once you know the new center and the radius, plug them into $(x-h)^2 + (y-k)^2 = r^2$ and compare your result to the answer choices.

In the first line, type (x-3)^2 + (y+4)^2 = 81 to see the original circle. Notice its center by hovering near the middle or visually estimating its position.

From the original center, imagine (or plot a point) 2 units to the right and 5 units down. In Desmos, you can add a point, for example (3,-4), then add another point (3+2, -4-5) and see where that new point lies.

Enter each option on a new line: (x+5)^2 + (y-2)^2 = 81, (x-5)^2 + (y+9)^2 = 81, (x-1)^2 + (y+1)^2 = 81, and (x+1)^2 + (y-9)^2 = 81. For each circle, look at its center and decide which one is centered exactly at the point that is 2 units right and 5 units down from the original center.

The general form of a circle’s equation is

where $(h,k)$ is the center and $r$ is the radius.

The given equation is $(x-3)^2 + (y+4)^2 = 81$.

• Compare $(x-3)$ with $(x-h)$, so $h = 3$.
• Compare $(y+4)$ with $(y-k)$, so $k = -4$ (because $y+4 = y-(-4)$).
• The right side is $81$, so $r^2 = 81$ and $r = 9$.

So the original circle has center $(3,-4)$ and radius $9$.

The circle is translated:

• 2 units to the right: this adds 2 to the $x$-coordinate of the center.
• 5 units downward: this subtracts 5 from the $y$-coordinate of the center.

Start with center $(3,-4)$:

• New $x$-coordinate: $3 + 2 = 5$.
• New $y$-coordinate: $-4 - 5 = -9$.

The new center is $(5,-9)$, and the radius is still $9$ because translations do not change size.

Using the form $(x-h)^2 + (y-k)^2 = r^2$:

• For center $(5,-9)$, we have $h = 5$ and $k = -9$.
• The radius is $9$, so $r^2 = 81$.

Substitute $h = 5$, $k = -9$, and $r^2 = 81$ into the formula to get the equation of the new circle.

With $h = 5$ and $k = -9$, the new circle’s equation is

This exactly matches answer choice B.