What are the methods for transforming rotated conic sections to standard form?

[notnottrue]

I am trying to find the level curves for the function g(x,y)= k = xy/(x^2+y^2).
I get, x^2+y^2-xy/k=0.
I know this is an ellipse, but I do not know how to factor, and find values of k for which the level curves exist.

 

[Sourabh N]

I am trying to find the level curves for the function g(x,y)= k = xy/(x^2+y^2).
I get, x^2+y^2-xy/k=0.
I know this is an ellipse, but I do not know how to factor, and find values of k for which the level curves exist.

If the level curves have to be ellipses, you can use the constraint on a general conic section (any quadratic equation in two variables) for it to be an ellipse. Applying the constraint to your quadratic equation will give you the values of k.

 

[HallsofIvy]

There are various ways to change "rotated" conic sections (that have an "xy" term) to "standard form" (without the xy). One is to rotate the coordinate system by writing $x= x'cos(\theta)+ y'sin(\theta)$, $y= -x'sin(\theta)+ y'cos(\theta)$. Put those into the equation and combine all x'y' terms. Choose $\theta$ to make its coefficient 0.

A method that is more 'advanced' but simple to compute is to write the second order terms as a matrix multiplication: write $ax^2+ bxy+ cy^2$ as
$$\begin{bmatrix}x & y \end{bmatrix}\begin{bmatrix}a & b/2 \\ b/2 & c\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}$$
and find the eigenvalues and eigenvectors of that matrix. The eigenvalues will be the coefficients of x² and y² and the eigenvectors will give the directions for the rotated coordinate system.

(I "distributed" the b as the b/2 off diagonal terms to make this a symmetric matrix which guarentees that it has real eigenvalues and independent eigenvectors.)