Example Question - circle radius and diameter

Finding the Diameter of a Circle from Its Expanded Equation

\begin{align*} &\text{Given equation: } x^2 + y^2 - 22x + 30y + 90 = 0 \\ &\text{Complete the square for the } x \text{ terms: } \\ &(x^2 - 22x) + y^2 + 30y + 90 = 0 \\ &(x^2 - 22x + 121) - 121 + y^2 + 30y + 90 = 0 \\ &(x - 11)^2 + y^2 + 30y + 90 - 121 = 0 \\ &\text{Complete the square for the } y \text{ terms: } \\ &(x - 11)^2 + (y^2 + 30y + 225) - 225 + 90 - 121 = 0 \\ &(x - 11)^2 + (y + 15)^2 = 256 \\ &\text{The standard form of a circle's equation is } (x - h)^2 + (y - k)^2 = r^2 \\ &\text{Comparing, we find the radius } r \text{ is } \sqrt{256} = 16. \\ &\text{The diameter } D \text{ is twice the radius: } D = 2r = 2 \times 16 = 32. \end{align*}