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Complex Numbers and Quadratic Equations · Question 2

JEE Main 2026 (Online) 6th April Morning ShiftMCQHard📈 Topic in 26% papers+4−1
1

Let e1 and e2 be two distinct roots of the equation x2-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 :

  1. 18
  2. 22
  3. 26
  4. 34

Answer

(3) 26

Solution

Let h(x)=x2-ax+2; product of roots =2>0, sum =a

Ellipse : 0<e<1, hyperbola : e>1

Case 1 : both roots distinct and greater than 1

(i) D>0 a2-8>0 |a|>2 2

(ii) h(1)>0 3-a>0 a<3

(iii) vertex to the right of 1 : a2>1 a>2

a (2 2, 3 ) =2 2, =3

Case 2 : one root in (0, 1) and the other greater than 1

h(0)=2>0 and h(1)<0 3-a<0 a>3

( , )=(3, ) =3

2+ 2+ 2=8+9+9=26

Answer: (3)

JEE Main 2026 (Online) 6th April Morning Shift

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