Let z1 and z2 be two complex numbers such that [ z _1 = i z _2] and [ ( z_1 z _2 ) = ].

Question

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 :

Accepted Answer

$$\because$$ $${{{z_1}} \over {{z_2}}} =  - i \Rightarrow {z_1} =  - i{z_2}$$

$$ \Rightarrow \arg ({z_1}) =  - {\pi  \over 2} + \arg ({z_2})$$ ..... (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 $$\arg ({z_2}) = {{3\pi } \over 4}$$

Solutions · 1