Question

the equation ( x^{2}+y^{2}+18x - 6y=-26 ) represents a circle. which equation could be used to find the center of the circle? a. ( (x + 9)^{2}+(y + 3)^{2}=64 ) b. ( (x + 9)^{2}+(y - 3)^{2}=64 ) c. ( (x + 81)^{2}+(y - 9)^{2}=64 ) d. ( (x + 81)^{2}+(y + 9)^{2}=64 )

Explanation:

Step1: Complete the square for \(x\)

For \(x^{2}+18x\), we have \((x + 9)^{2}=x^{2}+18x + 81\), so \(x^{2}+18x=(x + 9)^{2}-81\).

Step2: Complete the square for \(y\)

For \(y^{2}-6y\), we have \((y-3)^{2}=y^{2}-6y + 9\), so \(y^{2}-6y=(y - 3)^{2}-9\).

Step3: Rewrite the original equation

The original equation \(x^{2}+y^{2}+18x-6y=-26\) becomes \((x + 9)^{2}-81+(y - 3)^{2}-9=-26\).

Step4: Simplify the equation

\((x + 9)^{2}+(y - 3)^{2}-90=-26\), then \((x + 9)^{2}+(y - 3)^{2}=64\).

Answer:

B. \((x + 9)^{2}+(y - 3)^{2}=64\)