Problem:
Given the equation $-3 x^2+2 x+6 y^2-5 y+2=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Hyperbola
Equation: $6 \left(y-\frac{5}{12}\right)^2-3 \left(x-\frac{1}{3}\right)^2=-\frac{31}{24}$
Foci: $\left(
\begin{array}{cc}
 \frac{1}{12} \left(4-\sqrt{93}\right) & \frac{5}{12} \\
 \frac{1}{12} \left(4+\sqrt{93}\right) & \frac{5}{12} \\
\end{array}
\right)$
Eccentricity: $\sqrt{\frac{3}{2}}$
Center: $\left\{\frac{1}{2} \left(\frac{1}{12} \left(4-\sqrt{93}\right)+\frac{1}{12} \left(4+\sqrt{93}\right)\right),\frac{5}{12}\right\}$
Asymptotes: $\left\{y=\frac{x}{\sqrt{2}}+\frac{1}{12} \left(5-2 \sqrt{2}\right),y=\frac{1}{12} \left(5+2 \sqrt{2}\right)-\frac{x}{\sqrt{2}}\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/47830.txt'}