Question

give the center and radius of the circle represented by the equation.
25x² + 25y² - 30x + 30y - 31 = 0

the center of the circle is □.
(type an ordered pair, using integers or frac ons.)
the radius of the circle is □.
(type an integer or a simplified fraction.)

Explanation:

Step1: Divide the equation by 25

$$x^{2}+y^{2}-\frac{6}{5}x+\frac{6}{5}y-\frac{31}{25}=0$$

Step2: Rearrange the terms

$$x^{2}-\frac{6}{5}x + y^{2}+\frac{6}{5}y=\frac{31}{25}$$

Step3: Complete the square for x - terms

For \(x^{2}-\frac{6}{5}x\), we have \((x - \frac{3}{5})^{2}=x^{2}-\frac{6}{5}x+\frac{9}{25}\).

Step4: Complete the square for y - terms

For \(y^{2}+\frac{6}{5}y\), we have \((y+\frac{3}{5})^{2}=y^{2}+\frac{6}{5}y+\frac{9}{25}\).

Step5: Add the square - completed values to both sides

\((x - \frac{3}{5})^{2}- \frac{9}{25}+(y+\frac{3}{5})^{2}-\frac{9}{25}=\frac{31}{25}\)
\((x - \frac{3}{5})^{2}+(y+\frac{3}{5})^{2}=\frac{31 + 9+9}{25}\)
\((x - \frac{3}{5})^{2}+(y+\frac{3}{5})^{2}=\frac{49}{25}\)

Answer:

The center of the circle is \((\frac{3}{5},-\frac{3}{5})\).
The radius of the circle is \(\frac{7}{5}\).