JEE MAIN - Mathematics (2023 - 29th January Morning Shift - No. 10)
For two non-zero complex numbers $$z_{1}$$ and $$z_{2}$$, if $$\operatorname{Re}\left(z_{1} z_{2}\right)=0$$ and $$\operatorname{Re}\left(z_{1}+z_{2}\right)=0$$, 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 :
Explanation
Let, $${z_1} = {x_1} + i{y_1}$$
and $${z_2} = {x_2} + i{y_2}$$
$$\therefore$$ $${z_1}{z_2} = {x_1}{x_2} - {y_1}{y_2} + i({x_1}{y_2} + {x_2}{y_1})$$
Given, $${\mathop{\rm Re}\nolimits} ({z_1} + {z_2}) = 0$$
$$ \Rightarrow {x_1} + {x_2} = 0$$ ...... (1)
also given, $${\mathop{\rm Re}\nolimits} ({z_1}{z_2}) = 0$$
$$ \Rightarrow {x_1}{x_2} - {y_1}{y_2} = 0$$
$$ \Rightarrow {x_1}{x_2} = {y_1}{y_2}$$
$$ \Rightarrow {y_1}{y_2} = - x_1^2$$ [$$\because$$ $${x_2} = - {x_1}$$]
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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