Eccentricity of a general ellipse

Question

How to find the eccentricity of an ellipse $5x^2 + 5y^2 + 6xy = 8$ ?.

I tried it by factorizing it into the distance form for a line and point but I failed. Please help

Answer 1

Maybe not the quickest way to do it, but you can rotate it by $45^\circ$ with the substitution \begin{align} x&=\frac{X+Y}{\sqrt2}\\ y&=\frac{-X+Y}{\sqrt2}\,, \end{align} and as long as you know how to get the eccentricity of $a^2X^2+b^2Y^2=c^2$, you’re ready to go.

Answer 2

The eccentricity of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is $\sqrt{1-b^2/a^2}$. For an ellipse given by a quadratic equation $$ (x,y) V \pmatrix{x\cr y\cr} = 1$$ where $V$ is a positive definite symmetric matrix, the semi-major and semi-minor axes are the square roots of the reciprocals of the eigenvalues of $V$. In this case your matrix is $$ \pmatrix{5/8 & 3/8\cr 3/8 & 5/8\cr}$$ and its eigenvalues are $1$ and $1/4$.