Question

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

Explanation:

Step1: Rearrange the equation

Rearrange the given equation \(x^{2}+y^{2}-16 = 16y\) to \(x^{2}+y^{2}-16y-16=0\).

Step2: Complete the square for the \(y\) - terms

For the \(y\) - terms \(y^{2}-16y\), using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a = y\) and \(2ab=16y\) (so \(b = 8\)).
\(y^{2}-16y=(y - 8)^{2}-64\).
The equation becomes \(x^{2}+(y - 8)^{2}-64-16=0\).

Step3: Simplify the equation

Simplify \(x^{2}+(y - 8)^{2}-64-16=0\) to \(x^{2}+(y - 8)^{2}=80\).

Answer:

\(x^{2}+(y - 8)^{2}=80\)