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

◻

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

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Volume of cylinders

Full solution

Q. A circle in the

x y

-plane has the equation

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

. What is the diameter of the circle?

\newline

\square

Complete the square for x-terms: First, we need to complete the square for the x-terms in the equation. $\newline$ $4x^2 - 24x$ can be written as $4(x^2 - 6x)$ . $\newline$ To complete the square, we add $(6/2)^2 = 9$ to the expression inside the parenthesis. $\newline$ So, we have $4(x^2 - 6x + 9)$ .
Add to balance the equation: We also need to add $4\times 9$ to the other side of the equation to keep it balanced. $\newline$ The equation becomes $4(x^2 - 6x + 9) + 4y^2 = 28 + 4\times 9$ .
Simplify the equation: Now, simplify the equation. $\newline$ $4(x - 3)^2 + 4y^2 = 28 + 36$ . $\newline$ $4(x - 3)^2 + 4y^2 = 64$ .
Divide by $4$ for standard form: Divide the whole equation by $4$ to get the standard form of the circle's equation. $(x - 3)^2 + y^2 = 16$ .
Identify center and radius: 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. $\newline$ From our equation, $r^2 = 16$ , so $r = 4$ .
Calculate the diameter: The diameter of a circle is twice the radius. $\newline$ So, the diameter is $2 \times 4 = 8$ .

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