Question

give the center and radius of the circle represented by the equation.
8x² - 24x + 8y² + 72y + 52 = 0
the center of the circle is. (type an ordered pair, using integers or fractions.)
the radius of the circle is. (type an integer or a simplified fraction.)

Explanation:

Step1: Divide the equation by 8

$$x^{2}-3x + y^{2}+9y+\frac{13}{2}=0$$

Step2: Complete the square for x and y terms

For \(x\): \((x - \frac{3}{2})^{2}=x^{2}-3x+\frac{9}{4}\)
For \(y\): \((y + \frac{9}{2})^{2}=y^{2}+9y+\frac{81}{4}\)
The equation becomes \((x - \frac{3}{2})^{2}-\frac{9}{4}+(y + \frac{9}{2})^{2}-\frac{81}{4}+\frac{13}{2}=0\)

Step3: Simplify the equation

\((x - \frac{3}{2})^{2}+(y + \frac{9}{2})^{2}=\frac{9}{4}+\frac{81}{4}-\frac{13}{2}\)
\((x - \frac{3}{2})^{2}+(y + \frac{9}{2})^{2}=\frac{9 + 81-26}{4}=\frac{64}{4} = 16\)

Answer:

The center of the circle is \((\frac{3}{2},-\frac{9}{2})\).
The radius of the circle is \(4\).