A circle in the xy-plane has the equation x^(2)+y^(2)-22 x+30 y+90=0". " 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 - 15^2Isolate Constant Term: Now, add the squares and subtract the constants from both sides to isolate the constant term. (x - 11)^2 - 121 + (y + 15)^2 - 225 + 90 = 0 Combine like terms. (x - 11)^2 + (y + 15)^2 - 256 = 0Combine Like Terms: Add 256 to both sides to get the standard form of the circle's equation. (x - 11)^2 + (y + 15)^2 = 256Standard 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. Here, r^2 = 256, so r = √256. Calculate the radius. r = 16Calculate Radius: The diameter of a circle is twice the radius. Diameter = 2 * r Calculate the diameter. Diameter = 2 * 16Calculate Diameter: Finish the calculation to find the diameter. Diameter = 32

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Math Problems

Algebra 2

Find properties of circles from equations in general form

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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 = $32$