Where an ellipse is expressed in quadratic form (e.g. $ax^2 + bxy + cy^2 = k$ is expressed as $x^TQx = k$), the principal axes are in the directions given by the eigenvectors of Q. I understand this.
Now, if we want to find the intercepts of the ellipse and the principal axes, apparently they can be found as intercepts = $\pm \sqrt{k/\lambda_i}$ (where $\lambda_i$ is the eigenvalue corresponding to the eigenvector which describes the axis we are looking for the intercept with).
What I don't understand is how to derive this. I can see how to derive the intercepts in terms of $a, b, c, k$ but not in terms of the eigenvaluesand $k$.
Can someone please help shed some light on this - perhaps provide a derivation?
Thanks in advance for your help.