Question #40811

A family of conics $C_\lambda$ is defined by the matrix equation $X^T M_\lambda X = 0$, where $X = \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$ and $M_\lambda = \begin{pmatrix} \lambda & 1 & 0 \\ 1 & \lambda & 0 \\ 0 & 0 & -1 \end{pmatrix}$ The condition that a line $lx + my + n = 0$ is tangent to $C_\lambda$ is given by the vanishing of the bordered determinant: $\Delta(\lambda) = \begin{vmatrix} \lambda & 1 & 0 & l \\ 1 & \lambda & 0 & m \\ 0 & 0 & -1 & n \\ l & m & n & 0 \end{vmatrix} = 0$ A student simplifies $\Delta(\lambda)$ by performing the elementary row operation $R_1 \to \lambda R_1 - R_2$, yielding a new determinant $\Delta'(\lambda)$. The student incorrectly assumes $\Delta'(\lambda) = 0$ is the exact condition of tangency for all $\lambda \in \mathbb{R}$. Consequently, the student finds a specific value $\lambda_0 \in \mathbb{R}$ for which $\Delta'(\lambda_0) = 0$ identically for all $l, m, n$, falsely concluding that every line in the plane is tangent to $C_{\lambda_0}$. Find the equation of the true director circle of the conic $C_{\lambda_0}$.