Question

write the equation in standard form for the circle ( y^{2}=-x^{2}-18x + 19 ).

Explanation:

Step1: Rearrange the equation

Move all terms to one side: \(x^{2}+y^{2}+18x - 19=0\).

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

For the \(x\) terms \(x^{2}+18x\), using the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), where \(a=x\) and \(2b = 18\) (so \(b = 9\)). Then \(x^{2}+18x=(x + 9)^{2}-81\).
The equation becomes \((x + 9)^{2}-81+y^{2}-19=0\).

Step3: Simplify the equation

Combine the constant terms: \((x + 9)^{2}+y^{2}=81 + 19\).

Answer:

\((x + 9)^{2}+y^{2}=100\)