Let e_1 and e_2 be two distinct roots of the equation x^2 - ax + 2 = 0. Let the sets \ a R : e_1 and e_2 are the eccentricities of hyperbolas \ = ( , ), and \ a R : e_1 and e_2 are the eccentricities of an ellipse and a hyperbola, respectively \ = ( , ). Then ^2 + ^2 + ^2 is equal to:
C. 26
The given quadratic equation is x^2 - ax + 2 = 0 with roots e_1 and e_2. The sum of the roots is e_1 + e_2 = a and the product is e_1 e_2 = 2. For the roots to be real and distinct, the discriminant must be positive: = a^2 - 8 > 0 a (- , -2 2 ) (2 2 , ) Since e_1 and e_2 represent eccentricities, they must be positive. Given e_1 e_2 = 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 e_1 and e_2 to be the eccentricities of hyperbolas, both must be strictly greater than 1. This requires 1 to lie outside the roots and the vertex of the parabola to be to the right of 1: (e_1 - 1)(e_2 - 1) > 0 e_1 e_2 - (e_1 + e_2) + 1 > 0 2 - a + 1 > 0 a Combining this with the discriminant condition, we get a (2 2 , 3). Thus, = 2 2 and = 3. For e_1 and e_2 to be the eccentricities of an ellipse and a hyperbola, one root must be in (0, 1) and the other in (1, ). This requires 1 to lie between the roots: (e_1 - 1)(e_2 - 1) 3 For a > 3, the roots are real, distinct, and positive. Thus, the set of such a is (3, ). This gives = 3. Finally, calculating the required sum: ^2 + ^2 + ^2 = (2 2 )^2 + 3^2 + 3^2 = 8 + 9 + 9 = 26 Answer: 26
Related: Mathematics — Quadratic Equation · All PYQ Banks