Question

write the equation in standard form for the circle $y^{2}-11=-x^{2}-2x$.

Explanation:

Step1: Rearrange the equation

Move all terms to one side: $x^{2}+2x + y^{2}=11$.

Step2: Complete the square for the $x$ - terms

For the expression $x^{2}+2x$, using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$, where $a = x$ and $2ab=2x$ (so $b = 1$). Then $x^{2}+2x=(x + 1)^{2}-1$.
The equation becomes $(x + 1)^{2}-1+y^{2}=11$.

Step3: Simplify the equation

Add 1 to both sides of the equation: $(x + 1)^{2}+y^{2}=12$.

Answer:

$(x + 1)^{2}+y^{2}=12$