Let $e_{1}$ and $e_{2}$ be two distinct roots of the equation $x^{2} - ax + 2 = 0$. Let the sets$\le

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Let $e_{1}$ and $e_{2}$ be two distinct roots of the equation $x^{2} - ax + 2 = 0$. Let the sets

$\left\{ a \in \mathbb{R}:e_{1} \right.\ $ and $e_{2}$ are the eccentricities of hyperbolas $\} = (\alpha,\beta)$, and $\left\{ a \in \mathbb{R}:e_{1} \right.\ $ and $e_{2}$ are the eccentricities of an ellipse and a hyperbola, respectively $\} = (\gamma,\infty)$.

The $\alpha^{2} + \beta^{2} + \gamma^{2}$ is equal to:

Accepted Answer

Sol. [Key Concept: With $e_{1} + e_{2} = a,\ e_{1}e_{2} = 2$: the both-hyperbola case needs $e_{1},e_{2} > 1$ and the ellipse-plus-hyperbola case needs one $< 1$ and one $> 1$; each gives a range of $a$.]

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