Question:

Step 1: Write down the given equation of the circle. The equation of the circle provided is: $$x^2 + y^2 - 10x + 16y + 89 = r^2$$ Step 2: Rearrange the terms to group x-terms and y-terms together. Group the $x$-terms and $y$-terms to prepare for completing the square: $$(x^2 - 10x) + (y^2 + 16y) + 89 = r^2$$ Step 3: Complete the square for the x-terms. To complete the square for the $x$-terms ($x^2 - 10x$), we add and subtract $(\frac{-10}{2})^2 = (-5)^2 = 25$. $$(x^2 - 10x + 25 - 25) + (y^2 + 16y) + 89 = r^2$$ This can be rewritten as: $$(x-5)^2 - 25 + (y^2 + 16y) + 89 = r^2$$ Step 4: Complete the square for the y-terms. To complete the square for the $y$-terms ($y^2 + 16y$), we add and subtract $(\frac{16}{2})^2 = (8)^2 = 64$. $$(x-5)^2 - 25 + (y^2 + 16y + 64 - 64) + 89 = r^2$$ This can be rewritten as: $$(x-5)^2 - 25 + (y+8)^2 - 64 + 89 = r^2$$ Step 5: Simplify the equation into the standard form of a circle. Combine the constant terms on the left side: $$(x-5)^2 + (y+8)^2 - 25 - 64 + 89 = r^2$$ $$(x-5)^2 + (y+8)^2 - 89 + 89 = r^2$$ $$(x-5)^2 + (y+8)^2 = r^2$$ This is the standard form of the equation of a circle $(x-h)^2 + (y-k)^2 = R^2$. Step 6: Identify the center and radius of the circle. By comparing the derived standard form $(x-5)^2 + (y+8)^2 = r^2$ with the general standard form $(x-h)^2 + (y-k)^2 = R^2$: The center of the circle $(h, k)$ is $(5, -8)$. The square of the radius $R^2$ is $r^2$. Therefore, the radius $R$ is $\sqrt{r^2} = |r|$. Step 7: State the final answer. The center of the circle is $(5, -8)$ and the radius is $|r|$.