JEE MAIN - Mathematics (2022 - 25th June Evening Shift - No. 3)
Let z1 and z2 be two complex numbers such that $${\overline z _1} = i{\overline z _2}$$ and $$\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi $$. Then :
$$\arg {z_2} = {\pi \over 4}$$
$$\arg {z_2} = - {{3\pi } \over 4}$$
$$\arg {z_1} = {\pi \over 4}$$
$$\arg {z_1} = - {{3\pi } \over 4}$$
Explanation
$$\because$$ $${{{z_1}} \over {{z_2}}} = - i \Rightarrow {z_1} = - i{z_2}$$
$$ \Rightarrow \arg ({z_1}) = - {\pi \over 2} + \arg ({z_2})$$ ..... (i)
Also $$\arg ({z_1}) - \arg ({\overline z _2}) = \pi $$
$$ \Rightarrow \arg ({z_1}) + \arg ({z_2}) = \pi $$ ..... (ii)
From (i) and (ii), we get
$$\arg ({z_1}) = {\pi \over 4}$$ and $$\arg ({z_2}) = {{3\pi } \over 4}$$
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