Conic Sections · Eccentricity of ellipse and hyperbola

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)

. The

\alpha^2 + \beta^2 + \gamma^2

is equal to:

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