Problem:
Given the equation $2 x^2-5 x-8 y^2+4 y-1=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Hyperbola
Equation: $2 \left(x-\frac{5}{4}\right)^2-8 \left(y-\frac{1}{4}\right)^2=\frac{29}{8}$
Foci: $\left(
\begin{array}{cc}
 \frac{5}{4}-\frac{\sqrt{145}}{8} & \frac{1}{4} \\
 \frac{1}{8} \left(10+\sqrt{145}\right) & \frac{1}{4} \\
\end{array}
\right)$
Eccentricity: $\frac{\sqrt{5}}{2}$
Center: $\left\{\frac{1}{2} \left(\frac{5}{4}-\frac{\sqrt{145}}{8}+\frac{1}{8} \left(10+\sqrt{145}\right)\right),\frac{1}{4}\right\}$
Asymptotes: $\left\{y=\frac{x}{2}-\frac{3}{8},y=\frac{7}{8}-\frac{x}{2}\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/28328.txt'}