Problem:
Given the equation $10 x^2+8 x+5 y^2+10 y-9=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Ellipse
Equation: $10 \left(x+\frac{2}{5}\right)^2+5 (y+1)^2=\frac{78}{5}$
Foci: $\left(
\begin{array}{cc}
 -\frac{2}{5} & -1-\frac{\sqrt{39}}{5} \\
 -\frac{2}{5} & \frac{1}{5} \left(\sqrt{39}-5\right) \\
\end{array}
\right)$
Eccentricity: $\frac{1}{\sqrt{2}}$
Center: $\left\{-\frac{2}{5},\frac{1}{2} \left(-1-\frac{\sqrt{39}}{5}+\frac{1}{5} \left(\sqrt{39}-5\right)\right)\right\}$
Area Enclosed: $\frac{39 \sqrt{2} \pi }{25}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/4844.txt'}