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Question

Mathematics Question on complex numbers

Let z 1 and z 2 be two complex numbers such that

z1=iz2 and arg(z1z2)=π.z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.

A

arg z2=(π4)arg \,z_2=(\frac{\pi}{4})

B

arg z2=−(3π4)arg \,z_2=-(\frac{3\pi}{4})

C

arg z1=(π4)arg \,z_1=(\frac{\pi}{4})

D

arg z1=−(3π4)arg \,z_1=-(\frac{3\pi}{4})

Answer

arg z2=(π4)arg \,z_2=(\frac{\pi}{4})

Explanation

Solution

The correct option is: (C).

∵z1z2=i⇒z1=−iz2\frac{z_1}{z_2}=i⇒z_1=-iz_2

⇒arg(z1)=−π2+arg(z2).....(i)⇒arg(z_1)=-\frac{\pi}{2}+arg(z_2).....(i)

Also

arg(z1)-arg(z2)= π

⇒ arg(z 1) + arg(z 2) = π …(ii)

From (i) and (ii), we get

arg(z1)=π4and arg(z2)=3π4arg(z_1)=\frac{\pi}{4} and\,arg(z_2)=\frac{3\pi}{4}