JEE Main202310 Apr 2023Evening ShiftMathematicsComplex NumberActual
Let S = z = x + i y : 2 z - 3 i 4 z + 2 i is a real number . Then which of the following is NOT correct?
Options
- Ay + x 2 + y 2 ≠ - 1 4
- B( x , y ) = 0 , - 1 2
- Cx = 0
- Dy ∈ - ∞ , - 1 2 ∪ - 1 2 , ∞
Correct answer
B. ( x , y ) = 0 , - 1 2
Step-by-step solution
We have, Z = 2 z - 3 i 4 z + 2 i ⇒ Z = 2 x + i 2 y - 3 4 x + i 4 y + 2 ⇒ Z = 2 x + i 2 y - 3 4 x + i 4 y + 2 × 4 x - i 4 y + 2 4 x - i 4 y + 2 ⇒ Z = 8 x 2 + 4 y + 2 2 y - 3 + i 4 x 2 y - 3 - 8 x y + 4 x 16 x 2 + 4 y + 2 2 Since, Z is purely real, so 4 x 2 y - 3 - 8 x y + 4 x 16 x 2 + 4 y + 2 2 = 0 ⇒ - 12 x + 8 x y - 8 x y + 4 x = 0 ⇒ x = 0 And, 4 x + i 4 y + 2 ≠ 0 ⇒ i 4 y + 2 ≠ 0 ⇒ y ≠ - 1 2 Hence this is the required option.