Problem:
Given the equation $-6 x^2-x+y^2+4 y+2=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Hyperbola
Equation: $(y+2)^2-6 \left(x+\frac{1}{12}\right)^2=\frac{47}{24}$
Foci: $\left(
\begin{array}{cc}
 -\frac{1}{12} & -2-\frac{\sqrt{329}}{12} \\
 -\frac{1}{12} & \frac{\sqrt{329}}{12}-2 \\
\end{array}
\right)$
Eccentricity: $\sqrt{\frac{7}{6}}$
Center: $\left\{-\frac{1}{12},-2\right\}$
Asymptotes: $\left\{y=\frac{1}{12} \left(-24-\sqrt{6}\right)-\sqrt{6} x,y=\sqrt{6} x+\frac{1}{12} \left(\sqrt{6}-24\right)\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/26770.txt'}