Problem:
Given the equation $x^2+6 x+6 y^2-4 y-8=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Ellipse
Equation: $(x+3)^2+6 \left(y-\frac{1}{3}\right)^2=\frac{53}{3}$
Foci: $\left(
\begin{array}{cc}
 -3-\frac{\sqrt{\frac{265}{2}}}{3} & \frac{1}{3} \\
 \frac{1}{6} \left(\sqrt{530}-18\right) & \frac{1}{3} \\
\end{array}
\right)$
Eccentricity: $\sqrt{\frac{5}{6}}$
Center: $\left\{\frac{1}{2} \left(-3-\frac{\sqrt{\frac{265}{2}}}{3}+\frac{1}{6} \left(\sqrt{530}-18\right)\right),\frac{1}{3}\right\}$
Area Enclosed: $\frac{53 \pi }{3 \sqrt{6}}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/20648.txt'}