If (2 z1/3 z2) is a purely imaginary number, then |(z1-z2/z1+z2)|=

Question

If (2 z1/3 z2) is a purely imaginary number, then |(z1-z2/z1+z2)|= (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

As given, let (2 z1/3 z2)= iy or (z1/z2)=(3/2) iy, so that |(z1-z2/z1+z2)|=|((z1/z2)-1/(z1)z2+1)|=|((3/2) i y-1/(3)2 i y+1)|=|(1-(3/2) i y/1+(3)2 i y)|=1 ∵|z|=| barz|

Q. If 3z2​2z1​​ is a purely imaginary number, then ∣∣​z1​+z2​z1​−z2​​∣∣​=

As given, let 3z2​2z1​​=iy or z2​z1​​=23​ iy, so that ∣∣​z1​+z2​z1​−z2​​∣∣​=∣∣​z2​z1​​+1z2​z1​​−1​∣∣​=∣∣​23​iy+123​iy−1​∣∣​=∣∣​1+23​iy1−23​iy​∣∣​=1 {∵∣z∣=∣zˉ∣}

Q. If $\frac{2 z _{1}}{3 z _{2}}$ is a purely imaginary number, then $∣ ∣ \frac{z _{1} - z _{2}}{z _{1} + z _{2}} ∣ ∣ =$