Let `Z_1` and `Z_2` be any two non-zero complex number such that `3|z_1|= 4 |Z_2 |. If z =( 3z_1)/(2z_2)+(2z_2)/(3z_1)` then

Options

A. `|z|= 1/2 sqrt(17/2)`
B. Im(z)=0
C. Re(z) =0
D. `|z|= sqrt(5/2)`

Explanation

(*) Given `3|z_1|=4|z_2|rArr |z_1|/|z_2|=4/3 [because z_2 ne 0 rArr |z_2|ne 0 ]`
`therefore z_1/z_2=|z_1/z_2|e^(I theta) and z_2/z_1=|z_2/z_1|e^(-I theta)`
`[ because z= |z|( cos theta + I sin theta)=|z| e^(I theta)]`
`rArr z_1/z_2=4/3e^(i theta) and z_2/z_1=3/4 e^(i theta)`
`rArr 3/2 z_1/z_2=2e^(i theta) and 2/3z_2/z_1=2 e^(i theta)`
On adding these two, we get
`z=3/2 z_1/z_2+2e^(i theta)+1/2e^(i theta]`
`= 2 cos theta + 2i sin theta + 1/2 cos "" theta - 1/2 sin "" theta`
`[ because e^(pm i theta)=(cos theta pm i sin theta)]`
`=5/2 cos theta + 3/2 i sin theta`
`rArr |z|= sqrt((5/2)^2+(3/2)^2)=sqrt((34)/4)=sqrt(17)/(2)`
Note that z is nither purely imaginary and nor purely real.
'*' None of the options is correct