Questions: Decide whether or not the equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. 8 x^2+8 x+8 y^2-64 y-382=0 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The graph of the equation is a line. B. The graph of the equation is a point. C. The graph of the equation is a circle with center (Type an ordered pair.) The radius of the circle is . D. The graph is nonexistent.

Decide whether or not the equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. 8 x^2+8 x+8 y^2-64 y-382=0 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The graph of the equation is a line. B. The graph of the equation is a point. C. The graph of the equation is a circle with center (Type an ordered pair.) The radius of the circle is . D. The graph is nonexistent.
Transcript text: Decide whether or not the equation has a circle as its graph. If it does, give the center and the radius. If it does not, describe the graph. \[ 8 x^{2}+8 x+8 y^{2}-64 y-382=0 \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The graph of the equation is a line. B. The graph of the equation is a point. C. The graph of the equation is a circle with center $\square$. (Type an ordered pair.) $\square$ The radius of the circle is $\square$. D. The graph is nonexistent.
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Solution

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Solution Steps

Step 1: Check for Circle Conditions

To determine if the equation represents a circle, we check if \(A = C\) and \(B = 0\). In this case, since \(A = 8\), \(B = 0\), and \(C = 8\), the equation does represent a circle.

Step 2: Find the Center

The center \((h, k)\) of the circle can be found by transforming the equation into its standard form. By completing the square, we find the center at \((h, k) = (-\frac{D}{2A}, -\frac{E}{2C}) = (-0.5, 4)\).

Step 3: Calculate the Radius

The radius \(r\) of the circle is found by taking the square root of the constant term on the right side of the equation, which is obtained after completing the square. Thus, \(r = \sqrt{h^2 + k^2 - F} = 19.96\).

Final Answer:

The equation represents a circle with center \((-0.5, 4)\) and radius \(r = 19.96\).

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