Problem:
Given the equation $x^2-8 x-8 y^2-y-1=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Hyperbola
Equation: $(x-4)^2-8 \left(y+\frac{1}{16}\right)^2=\frac{543}{32}$
Foci: $\left(
\begin{array}{cc}
 4-\frac{3 \sqrt{543}}{16} & -\frac{1}{16} \\
 4+\frac{3 \sqrt{543}}{16} & -\frac{1}{16} \\
\end{array}
\right)$
Eccentricity: $\frac{3}{2 \sqrt{2}}$
Center: $\left\{4,-\frac{1}{16}\right\}$
Asymptotes: $\left\{y=\frac{x}{2 \sqrt{2}}-\sqrt{2}-\frac{1}{16},y=\frac{1}{16} \left(16 \sqrt{2}-1\right)-\frac{x}{2 \sqrt{2}}\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/9940.txt'}