Verified Solution • 2025–26 Session moderate

QJEEM23Mae330: JEE Main 2023 Mathematics - Class 11 JEEM23Mae330 Solution: For two non-zero complex numbers and , if and , then which of…

For two non-zero complex numbers z1z_{1} and z2z_{2} , if Re⁡(z1z2)=0\operatorname{Re}\left(z_{1} z_{2}\right)=0 and Re⁡(z1+z2)=0\operatorname{Re}\left(z_{1}+z_{2}\right)=0 , then which of the following are possible?

A. \operatorname{Im}\left(z_{1}\right)>0 and \operatorname{Im}\left(z_{2}\right) > 0

B. \operatorname{Im}\left(z_{1}\right) < 0 and \operatorname{Im}\left(z_{2}\right) > 0

C. \operatorname{Im}\left(z_{1}\right) > 0 and \operatorname{Im}\left(z_{2}\right) < 0

D. \operatorname{Im}\left(z_{1}\right) < 0 and \operatorname{Im}\left(z_{2}\right) < 0

Choose the correct answer from the options given below :

Multiple Choice Options

  • A A and C
  • B A and B
  • C B and D
  • D B and C

Step-by-Step Verified Solution

Let, z1=x1+iy1{z_1} = {x_1} + i{y_1}

and z2=x2+iy2{z_2} = {x_2} + i{y_2}

∴\therefore z1z2=x1x2−y1y2+i(x1y2+x2y1){z_1}{z_2} = {x_1}{x_2} - {y_1}{y_2} + i({x_1}{y_2} + {x_2}{y_1})

Given, Re(z1+z2)=0{\mathop{\rm Re}\nolimits} ({z_1} + {z_2}) = 0

⇒x1+x2=0\Rightarrow {x_1} + {x_2} = 0 ...... (1)

also given, Re(z1z2)=0{\mathop{\rm Re}\nolimits} ({z_1}{z_2}) = 0

⇒x1x2−y1y2=0\Rightarrow {x_1}{x_2} - {y_1}{y_2} = 0

⇒x1x2=y1y2\Rightarrow {x_1}{x_2} = {y_1}{y_2}

⇒y1y2=−x12\Rightarrow {y_1}{y_2} = - x_1^2 [ ∵\because x2=−x1{x_2} = - {x_1} ]

So, multiplication of imaginary part's of z1 and z2 is negative. It means sign of y1 and y2 are opposite of each other.

∴\therefore B and C are correct.

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