<s>[INST] Area enclosed by the curve $5x^2+6xy+2y^2+7x+6y+6=0$ We have to find the area enclosed by the curve
$$5x^2+6xy+2y^2+7x+6y+6=0.$$
I tried and I got that it is an ellipse, and I know its area is $\pi ab$ where $a$ and $b$ are the semiaxis lengths of the ellipse.
But I am unable to find the value of $a$ and $b$.
 [/INST] For the curve $ax^2+2hxy+by^2+2gx+2fy+c = 0$, the center is given by 
\begin{align*}
ax+hy+g &= 0\\
hx+by+f &= 0
\end{align*}
and the equation of the ellipse with the axes through the center parallel to the axes is 
\begin{align*}
ax^2+2hxy+by^2 + c' = 0
\end{align*}
where $c' = gx_c + fy_c + c$, where $(x_c, y_c)$ is the center.
</s>