JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

For two non-zero complex numbers $z_1$ and $z_2$ , if $Re(z_1z_2) = 0$ 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:

Explanation

Solution

Let $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$ .

Given $Re(z_1 + z_2) = 0 \implies x_1 + x_2 = 0 \implies x_2 = -x_1$ .

Since $z_1, z_2$ are non-zero, let $x_1 = a$ (where $a \ne 0$ ). Then $x_2 = -a$ .

Now, $z_1z_2 = (a + iy_1)(-a + iy_2) = (-a^2 - y_1y_2) + i(ay_2 - ay_1)$ .

Given $Re(z_1z_2) = 0$ :

-a^2 - y_1y_2 = 0 \implies y_1y_2 = -a^2

Since $a^2$ is always positive (as $a \ne 0$ ), then $-a^2$ must be negative.

y_1y_2 < 0

This means $y_1$ and $y_2$ must have opposite signs.

If $y_1 > 0$ , then $y_2 < 0$ (Condition C)
If $y_1 < 0$ , then $y_2 > 0$ (Condition B)

Thus, B and C are the possible cases.

Correct Option: (3)