A circle in the $xy$ -plane has the equation $4x^2+4y^2-24x=28$ . What is the diameter of the circle?

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Convert equations of circles from general to standard form

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Q. A circle in the

xy

-plane has the equation

4x^2+4y^2-24x=28

. What is the diameter of the circle?

Write Circle Equation: Write the given equation of the circle. $\newline$ The given equation is $4x^2 + 4y^2 - 24x = 28$ . $\newline$ We need to complete the square for both $x$ and $y$ to bring the equation into standard form.
Factor Out Coefficients: Factor out the coefficient of $x^2$ and $y^2$ . $\newline$ Since the coefficients of $x^2$ and $y^2$ are both $4$ , we factor them out to make completing the square easier. $\newline$ $4(x^2 + y^2 - 6x) = 28$
Complete Square for $x$ : Complete the square for the $x$ -term. $\newline$ To complete the square, we take the coefficient of $x$ , divide it by $2$ , and square it. Then we add and subtract this value inside the parenthesis. $\newline$ $4(x^2 - 6x + 9 - 9 + y^2) = 28$
Add Value to Equation: Add the value we subtracted to the other side of the equation. $\newline$ We added and subtracted $9$ inside the parenthesis, but since it's multiplied by $4$ , we need to add $4\times9$ to the other side of the equation. $\newline$ $4(x^2 - 6x + 9 + y^2) = 28 + 4\times9$ $\newline$ $4(x^2 - 6x + 9 + y^2) = 28 + 36$ $\newline$ $4(x^2 - 6x + 9 + y^2) = 64$
Write Completed Square: Write the completed square for the $x$ -term and simplify the equation. $(x - 3)^2$ is the completed square for the $x$ -term, and we simplify the right side of the equation. $4((x - 3)^2 + y^2) = 64$
Divide by Coefficient: Divide both sides by the coefficient of the completed squares to get the standard form of the circle's equation. $\newline$ $(x - 3)^2 + y^2 = \frac{64}{4}$ $\newline$ $(x - 3)^2 + y^2 = 16$
Identify Circle Radius: Identify the radius of the circle from the standard form equation. $\newline$ 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. From our equation, $r^2 = 16$ , so $r = 4$ .
Calculate Circle Diameter: Calculate the diameter of the circle. $\newline$ The diameter $D$ of a circle is twice the radius $r$ , so $D = 2r$ . $\newline$ $D = 2 \times 4$ $\newline$ $D = 8$