How to convert the general form of ellipse equation to the standard form? $$-x+2y+x^2+xy+y^2=0$$ Thank you in advance?
The idea is to rotate your axes , so that the $xy$ term disappears. Define a transformation ( a rotation ) $x'=rcos\theta, y'=rsin\theta$ , sub-in in your equation, and set the mixed $x'y'$-terms equal to $0$. This will give you the necessary angle of rotation to make the $xy$ terms disappear.
See this post: How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form for another example.