JEE Main202329 Jan 2023Morning ShiftMathematicsComplex NumberActual
For two non-zero complex number z 1 and z 2 , if Re z 1 z 2 = 0 and Re z 1 + z 2 = 0 , then which of the following are possible? (A) Im z 1 > 0 and Im z 2 > 0 (B) Im z 1 < 0 and Im z 2 > 0 (C) Im z 1 > 0 and Im z 2 < 0 (D) Im z 1 < 0 and Im z 2 < 0 Choose the correct answer from the options given below:
Options
- AB and D
- BB and C
- CA and B
- DA and C
Correct answer
B. B and C
Step-by-step solution
Let, z 1 = x 1 + i y 1 and z 2 = x 2 + i y 2 ∵ Re z 1 z 2 = 0 x 1 x 2 - y 1 y 2 = 0 . . . . . . i ∵ Re z 1 + z 2 = 0 x 1 + x 2 = 0 . . . . . . i i From equations i and i i we get x 1 2 + y 1 y 2 = 0 ⇒ y 1 y 2 = - x 1 2 Therefore I m ( z 1 ) and I m ( z 2 ) are of opposite sign