In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$. Let $U$ be one of the points of $\mathcal{C}(\Omega, r)$ at which the tangent line contains $K$. Express the eigenvalues of $A$, considered as an element of $\mathcal{M}_2(\mathbb{C})$, using the abscissa of $K$ and the distance $KU$ from $K$ to $U$.
In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r \geqslant 0$ disjoint from the $x$-axis. We denote by $K$ the orthogonal projection of $\Omega$ onto the $x$-axis. Let $A$ be a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$. Let $U$ be one of the points of $\mathcal{C}(\Omega, r)$ at which the tangent line contains $K$.
Express the eigenvalues of $A$, considered as an element of $\mathcal{M}_2(\mathbb{C})$, using the abscissa of $K$ and the distance $KU$ from $K$ to $U$.