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 (C) Im (

Correct answer: B. B and C 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 . . . . . . (ii) From equations (i) and (ii) we get x _ 1 ^ 2 + y _ 1 y _ 2 = 0 ⇒ y _ 1 y _ 2 = - x _ 1 ^ 2 Therefore Im ( z _ 1 ) and Im ( z _ 2 ) are of opposite sign

JEE Main 202329 JanuaryMorning Shift

Exam: JEE Main
Year: 2023
Date: 29 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Complex Number

JEE Main 2023 · 29 January · Morning Shift · Mathematics · Complex Number

For two non-zero complex number z _ 1 and z _ 2 , if Re ( z _ 1 z _...

For two non-zero complex number ${z}_{1}$ and ${z}_{2}$ , if $Re \left({z}_{1}{z}_{2}\right) = 0$ and $Re \left({z}_{1} + {z}_{2}\right) = 0$ , then which of the following are possible?

(A) $Im \left({z}_{1}\right) > 0$ and $Im \left({z}_{2}\right) > 0$
(B) $Im \left({z}_{1}\right) < 0$ and $Im \left({z}_{2}\right) > 0$
(C) $Im \left({z}_{1}\right) > 0$ and $Im \left({z}_{2}\right) < 0$
(D) $Im \left({z}_{1}\right) < 0$ and $Im \left({z}_{2}\right) < 0$

Choose the correct answer from the options given below:

Options

A $B$ and $D$
B $B$ and $C$
C $A$ and $B$
D $A$ and $C$

Correct answer

B. $B$ and $C$

Step-by-step solution

Let,

${z}_{1} = {x}_{1} + i{y}_{1} and {z}_{2} = {x}_{2} + i{y}_{2}$

$∵ Re \left({z}_{1}{z}_{2}\right) = 0$

${x}_{1}{x}_{2} - {y}_{1}{y}_{2} = 0 . . . . . . \left(i\right)$

$∵ Re \left({z}_{1} + {z}_{2}\right) = 0$

${x}_{1} + {x}_{2} = 0 . . . . . . \left(ii\right)$

From equations \left(i\right) $and \left(ii\right)$ we get

${{x}_{1}}^{2} + {y}_{1}{y}_{2} = 0$

$⇒ {y}_{1}{y}_{2} = - {{x}_{1}}^{2}$

Therefore $Im ( {z}_{1} ) and Im ( {z}_{2} )$ are of opposite sign

Practice Complex Number on Quantrex Academy →

More from Complex Number