Problem:
Given the equation $-2 x^2+2 x+8 y^2-9 y+3=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Hyperbola
Equation: $8 \left(y-\frac{9}{16}\right)^2-2 \left(x-\frac{1}{2}\right)^2=-\frac{31}{32}$
Foci: $\left(
\begin{array}{cc}
 \frac{1}{16} \left(8-\sqrt{155}\right) & \frac{9}{16} \\
 \frac{1}{16} \left(8+\sqrt{155}\right) & \frac{9}{16} \\
\end{array}
\right)$
Eccentricity: $\frac{\sqrt{5}}{2}$
Center: $\left\{\frac{1}{2} \left(\frac{1}{16} \left(8-\sqrt{155}\right)+\frac{1}{16} \left(8+\sqrt{155}\right)\right),\frac{9}{16}\right\}$
Asymptotes: $\left\{y=\frac{x}{2}+\frac{5}{16},y=\frac{13}{16}-\frac{x}{2}\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/42001.txt'}