Problem:
Given the equation $x^2+7 x+5 y^2-y-9=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Ellipse
Equation: $\left(x+\frac{7}{2}\right)^2+5 \left(y-\frac{1}{10}\right)^2=\frac{213}{10}$
Foci: $\left(
\begin{array}{cc}
 -\frac{7}{2}-\frac{\sqrt{426}}{5} & \frac{1}{10} \\
 \frac{\sqrt{426}}{5}-\frac{7}{2} & \frac{1}{10} \\
\end{array}
\right)$
Eccentricity: $\frac{2}{\sqrt{5}}$
Center: $\left\{-\frac{7}{2},\frac{1}{10}\right\}$
Area Enclosed: $\frac{213 \pi }{10 \sqrt{5}}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/18330.txt'}