Coordinate Geometry: JEE Main 2026 Mathematics question

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-ax+2=0$. Let the sets $\{a\in\mathbb{R}: e_1$ and $e_2$ are the eccentricities of hyperbolas$\}=(\alpha,\beta)$, and $\{a\in\mathbb{R}: e_1$ and $e_2$ are the eccentricities of an ellipse and a hyperbola, respectively$\}=(\gamma,\infty)$. Then $\alpha^2+\beta^2+\gamma^2$ is equal to: