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Complex Numbers and Quadratic Equations

JEE Main 2026 (Online) 8th April Evening ShiftMCQMedium📈 Topic in 42% papers+4−1
Ques. 1JEE Main 2026, 8 April (Evening Shift)

The number of values of z C, satisfying the equations |z-(4+8i)|= 10 and |z-(3+5i)|+|z-(5+11i)|=4 5, is :

  1. 0
  2. 2
  3. 1
  4. 4

Answer

(2) 2

Solution

[figure]

|z-(3+5i)|+|z-(5+11i)|=4 5: sum of distances from S(3, 5) and S'(5, 11) is constant

ellipse with foci S, S' and 2a=4 5 a2=20

Centre C = mid point of SS' =(4, 8)

2c=SS'= 22+62=2 10 c2=10

b2=a2-c2=20-10=10 b= 10

|z-(4+8i)|= 10: circle with centre C and radius 10=b

every point of an ellipse is at a distance b from its centre, with equality only at the ends of the minor axis

the circle touches the ellipse only at the two ends of the minor axis, (1, 9) and (7, 7)

number of values of z =2

Answer: (2)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2026, 6 April (Morning Shift)

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
Ques. 3JEE Main 2026, 6 April (Morning Shift)

Let the set of all values of k R such that the equation z ( z+2+i )+k(2+3i)=0, z C, has at least one solution, be the interval [ , ]. Then 9( + ) is equal to :

  1. -10
  2. -8
  3. 10 13
  4. 8 13
Ques. 4JEE Main 2026, 6 April (Evening Shift)

Let S= \z C : z2+ 6 iz-3=0 \. Then z S z8 is equal to :

  1. 162
  2. 184
  3. 262
  4. 324
Ques. 5JEE Main 2026, 5 April (Morning Shift)

Let a, b C. Let , be the roots of the equation x2+ax+b=0. If - = 11 and 2- 2=3i 11, then ( 3- 3 )2 is equal to:

  1. 160
  2. 176
  3. 194
  4. 187
Ques. 6JEE Main 2026, 5 April (Evening Shift)

Let , be the roots of the equation x2-x+p=0 and , be the roots of the equation x2-4x+q=0; p, q Z. If , , , are in G.P., then |p+q| equals :

  1. 16
  2. 32
  3. 34
  4. 38
Ques. 7JEE Main 2026, 5 April (Evening Shift)

Let z1, z2 C be the distinct solutions of the equation z2+4z-(1+12i)=0. Then |z1|2+|z2|2 is equal to :

  1. 18
  2. 22
  3. 29
  4. 34
Ques. 8JEE Main 2026, 4 April (Morning Shift)

If the set of all solutions of |x2+x-9|=|x|+|x2-9| is [ , ] [ , ), then ( 2+ 2+ 2 ) is equal to:

  1. 9
  2. 18
  3. 36
  4. 72
Ques. 9JEE Main 2026, 4 April (Morning Shift)

Let z be a complex number such that |z+2|=|z-2| and ( z+3z-i )= 4. Then |z|2 is equal to:

  1. 9
  2. 4
  3. 5
  4. 1
Ques. 10JEE Main 2026, 4 April (Evening Shift)

Let S=\z C : z2+4z+16=0\. Then z S |z+ 3 i |2 is equal to :

  1. 42
  2. 23
  3. 27
  4. 38
Ques. 11JEE Main 2026, 4 April (Evening Shift)

If the quadratic equation ( +2)x2-3 x+4 =0, -2, has two positive roots, then the number of possible integral values of is :

  1. 1
  2. 2
  3. 3
  4. 4

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