Let e₁ and e₂ be two distinct roots of the equation x^2 - ax + 2 = 0 . Let the sets a R : e₁ and e₂ are the eccentricities of hyperbolas = ( , ) , and a R : e₁ and e₂ are the eccentricities of an ellipse and a hyperbola, respectively = ( , ) . Then ^2 + ^2 + ^2 is equal to:

Step-by-step solution

The given quadratic equation is x^2 - ax + 2 = 0 with roots e₁ and e₂ . The sum of the roots is e₁ + e₂ = a and the product is e₁ e₂ = 2 . For the roots to be real and distinct, the discriminant must be positive: = a^2 - 8 > 0 a (- , -2 2 ) (2 2 , ) Since e₁ and e₂ represent eccentricities, they must be positive. Given e₁ e₂ = 2 > 0 , both roots have the same sign. For them to be positive, their sum must be positive, so a > 0 . Thus, a > 2 2 . For e₁ and e₂ to be the eccentricities of hyperbolas, both must be stric