Question #41546

Let $C_1$ be the locus of points $(x,y)$ for $x > 0$ such that the matrix equation $\begin{pmatrix} x & 1 \\ x^2 & x \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} y \\ x+2 \end{pmatrix}$ has at least one solution for $(u,v)$. Let $C_2$ be the locus of points $(x,y)$ for $x > 0$ such that the matrix $M = \begin{pmatrix} y & 2 \\ 2 & x \end{pmatrix}$ has $\lambda = 0$ as an eigenvalue. If $R$ is the region in the first quadrant bounded by $C_1$, $C_2$, and the line $x = 1$, find the area of $R$.