Question - Finding the Diameter of a Circle from Its Expanded Equation
mathcompleting the square methodcircle diameter calculationexpanded circle equationcircle radius and diameter
Solution:
\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*}
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