Question

write the equation in standard form for the circle ( x^2 + y^2 - 16x - 32 = 0 ).

Explanation:

Step1: Group x-terms and move constant

Group the \(x\)-terms and move the constant to the right: \(x^{2}-16x + y^{2}=32\).

Step2: Complete the square for x

For \(x^{2}-16x\), take half of \(-16\) (which is \(-8\)), square it (\(64\)), and add to both sides: \(x^{2}-16x + 64 + y^{2}=32 + 64\).

Step3: Write as perfect squares

Rewrite \(x^{2}-16x + 64\) as \((x - 8)^{2}\), so the equation becomes \((x - 8)^{2}+y^{2}=96\).

Answer:

\((x - 8)^{2}+y^{2}=96\)