Question

rewrite the equation of the circle (x^{2}+y^{2}-8x + 12y+16 = 0) into standard form. enter your answers in the boxes to complete the equation. be sure to include an addition or subtraction symbol where needed. ((xsquare)^{2}+(ysquare)^{2}=square)

Explanation:

Step1: Group x - terms and y - terms

Given equation is \(x^{2}+y^{2}-8x + 12y+16 = 0\). Group the \(x\) - terms and \(y\) - terms: \((x^{2}-8x)+(y^{2}+12y)+16 = 0\).

Step2: Complete the square for x - terms

For the \(x\) - terms \(x^{2}-8x\), take half of the coefficient of \(x\) (\(\frac{-8}{2}=-4\)), then square it (\((-4)^{2}=16\)). Add and subtract 16 inside the first parenthesis: \((x^{2}-8x + 16-16)+(y^{2}+12y)+16 = 0\), which can be rewritten as \((x - 4)^{2}-16+(y^{2}+12y)+16 = 0\).

Step3: Complete the square for y - terms

For the \(y\) - terms \(y^{2}+12y\), take half of the coefficient of \(y\) (\(\frac{12}{2}=6\)), then square it (\(6^{2}=36\)). Add and subtract 36 inside the second parenthesis: \((x - 4)^{2}-16+(y^{2}+12y + 36-36)+16 = 0\), which can be rewritten as \((x - 4)^{2}+(y + 6)^{2}-16-36 + 16=0\).

Step4: Simplify the equation

Simplify the right - hand side: \((x - 4)^{2}+(y + 6)^{2}-36 = 0\), then \((x - 4)^{2}+(y+6)^{2}=36\).

Answer:

\((x - 4)^{2}+(y + 6)^{2}=36\)