For two non-zero complex number \(\mathrm{z}_{1}\) and \(\mathrm{z}_{2}\), if \(\operatorname{Re}(\mathrm{z}_{1} \mathrm{z}_{2})=0\) and \(\operatorname{Re}(\mathrm{z}_{1}+\mathrm{z}_{2})=0\), then which of the following are possible ? (A) \(\operatorname{Im}(\mathrm{z}_{1})>0\) and \(\operatorname{Im}(\mathrm{z}_{2})>0\) (B) \(\operatorname{Im}(\mathrm{z}_{1}) 0\) (C) \(\operatorname{Im}(\mathrm{z}_{1})>0\) and \(\operatorname{Im}(\mathrm{z}_{2})<0\) (D) \(\operatorname{Im}(\mathrm{z}_{1})<0\) and \(\operatorname{Im}(\mathrm{z}_{2})<0\) Choose the correct answer from the options given below :

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Complex Numbers

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 :

Answer: (B)

Put

z_{1} = x_{1} + i y_{1}

and

z_{2} = x_{2} + i y_{2}

. Then

Re (z_{1} z_{2}) = x_{1} x_{2} - y_{1} y_{2} = 0

gives

x_{1} x_{2} = y_{1} y_{2}

, and

Re (z_{1} + z_{2}) = x_{1} + x_{2} = 0

gives

x_{2} = - x_{1}

. For non-zero numbers

x_{1} \neq = 0

, so

x_{1} x_{2} = - x_{1}^{2} < 0

, and therefore

y_{1} y_{2} = x_{1} x_{2} < 0

: the imaginary parts have opposite signs. This excludes both parts positive and both parts negative, and allows exactly the cases

Im (z_{1}) < 0

with

Im (z_{2}) > 0

, and

Im (z_{1}) > 0

with

Im (z_{2}) < 0

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Single correct option