<s>[INST] Eccentricity of conic $4x^2+4xy+4y^2+x-5=0$ is 
Finding Eccentricity of conic $4x^2+4xy+4y^2+x-5=0$ is 

what I tried:
let $S = 4x^2+4xy+4y^2+x-5$
$\dfrac{dS}{dx}=8x+4y+1$ and $\dfrac{dS}{dy}=4x+8y$
for center $\dfrac{dS}{dx}=0$ and $\dfrac{dS}{dy}=0$
getting center as $ x=-\dfrac{1}{6}$ and $y=\dfrac{1}{12}$
How do I solve it? Help me please
 [/INST] The eccentricity only depends on the ratio of the axis so that only the quadratic terms matter.
By symmetry, it is obvious that one of the axis is parallel to $x=y$ and we apply the orthogonal transformation
$$x=u-v,y=u+v$$
which yields
$$3u^2+v^2.$$
Hence
$$e=\sqrt{1-\dfrac13}.$$

More generally, when the quadratic terms are $ax^2+2bxy+cy^2$, we compute the ratio of the Eigenvalues of
$$\begin{pmatrix}a&b\\b&c\end{pmatrix}$$ which are the roots of 
$$x^2-(a+c)x+ac-b^2,$$
and that ratio is
$$\dfrac{a+c-\sqrt{(a-c)^2+4b^2}}{a+c+\sqrt{(a-c)^2+4b^2}}.$$
</s>