Question

write the equation in standard form for the circle $y^{2}=-x^{2}+16x + 36$.

Explanation:

Step1: Rearrange the equation

Move all terms to one side: \(x^{2}+y^{2}-16x - 36=0\).

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

For the \(x\) - terms \(x^{2}-16x\), using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), where \(a = x\) and \(2b=16\) (so \(b = 8\)). Then \(x^{2}-16x=(x - 8)^{2}-64\).
The equation becomes \((x - 8)^{2}-64+y^{2}-36=0\).

Step3: Simplify the equation

\((x - 8)^{2}+y^{2}=64 + 36\).

Answer:

\((x - 8)^{2}+y^{2}=100\)