Question

complete the square to rewrite the equation of the circle in standard form. what is the equation of the circle in standard form? (x^{2}+y^{2}+4x - 10y + 20 = 0) ((x - 2)^{2}+(y + 5)^{2}=9) ((x + 2)^{2}+(y - 5)^{2}=9) ((x + 4)^{2}+(y - 10)^{2}=20) ((x - 4)^{2}+(y + 10)^{2}=20)

Explanation:

Step1: Group x - terms and y - terms

\(x^{2}+4x + y^{2}-10y+20 = 0\)

Step2: Complete the square for x - terms

For \(x^{2}+4x\), we add \((\frac{4}{2})^{2}=4\) to both sides of the equation. The equation becomes \(x^{2}+4x + 4+y^{2}-10y+20=4\).

Step3: Complete the square for y - terms

For \(y^{2}-10y\), we add \((\frac{-10}{2})^{2}=25\) to both sides of the equation. So \(x^{2}+4x + 4+y^{2}-10y+25+20=4 + 25\).

Step4: Rewrite in standard form

\((x + 2)^{2}+(y - 5)^{2}=9\)

Answer:

\((x + 2)^{2}+(y - 5)^{2}=9\)