Problem:
Given the equation $3 x^2+y^2+y-10=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Ellipse
Equation: $3 x^2+\left(y+\frac{1}{2}\right)^2=\frac{41}{4}$
Foci: $\left(
\begin{array}{cc}
 0 & -\frac{1}{2}-\sqrt{\frac{41}{6}} \\
 0 & \sqrt{\frac{41}{6}}-\frac{1}{2} \\
\end{array}
\right)$
Eccentricity: $\sqrt{\frac{2}{3}}$
Center: $\left\{0,-\frac{1}{2}\right\}$
Area Enclosed: $\frac{41 \pi }{4 \sqrt{3}}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/22893.txt'}