A circle in the $x y$ -plane has the equation $\newline$ $x^{2}+y^{2}-22 x+30 y+90=0 \text {. }$ $\newline$ How long is the diameter of the circle?

Full solution

Q. A circle in the

x y

-plane has the equation

\newline

x^{2}+y^{2}-22 x+30 y+90=0 \text {. }

\newline

How long is the diameter of the circle?

Group and Factor $y$ Terms: For $y$ : Group $y$ terms and factor out the coefficient of $y^2$ , which is $1$ . $y^2 + 30y = (y + 15)^2 - 15^2$
Isolate Constant Term: Now, add the squares and subtract the constants from both sides to isolate the constant term. $\newline$ $(x - 11)^2 - 121 + (y + 15)^2 - 225 + 90 = 0$ $\newline$ Combine like terms. $\newline$ $(x - 11)^2 + (y + 15)^2 - 256 = 0$
Combine Like Terms: Add $256$ to both sides to get the standard form of the circle's equation. $\newline$ $(x - 11)^2 + (y + 15)^2 = 256$
Standard Form of Circle's Equation: The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center and $r$ is the radius. $\newline$ Here, $r^2 = 256$ , so $r = \sqrt{256}$ . $\newline$ Calculate the radius. $\newline$ $r = 16$
Calculate Radius: The diameter of a circle is twice the radius. $\newline$ Diameter $= 2 \times r$ $\newline$ Calculate the diameter. $\newline$ Diameter $= 2 \times 16$
Calculate Diameter: Finish the calculation to find the diameter. $\newline$ Diameter = $32$