I have an equation of an ellipse (we know that this is an ellipse): $$7x^2 -4xy + 4y^2-6x – 12y = 9. $$
How do we determine, using linear algebra, the equation of the largest circle inscribed in this ellipse (by "largest", I mean with the highest radius).
I have no idea how to approach this. I can try and write simplify this equation by writing the equation in terms of matrices, like $$\begin{pmatrix} x & y \end{pmatrix} \cdot \begin{pmatrix}7 & -2 \\ -2 & 4 \end{pmatrix} \cdot \begin{pmatrix}x \\ y \end{pmatrix} + \begin{pmatrix} -6 & -12 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = 9, $$ and then write the equation in term of coordinates of the basis $\alpha$ or eigenvectors of the symmetric matrix from the left side of the equation, but i have no idea how to use this to determine the equation of a circle.
Best Answer
You're almost there! If you normalise your equation so that the constant on the right-hand side is $1$, then the major and minor radii of the ellipse are of the form $1/\sqrt \lambda$, where $\lambda$ is an eigenvalue of the matrix. So take the smaller of these as the radius of your circle.