Let a and b be two positive real numbers and `z_(1)` and `z_(2)` be two non-zero complex numbers such that `a|z_(1)|=b|z_(2)|`. If `z=(az_(1))/(bz_(2))+(bz_(2))/(az_(1))`, then

Options

A. Re(z)=0
B. Im(z)=0
C. |z|`=a/b`
D. `|z| gt 2`

Answer

B

Explanation

Let `omega=(az_(1))/(bz_(2))`. Then, `|omega|-|(az_(1))/(bz_(2))|=(a|z_(1)|)/(b|z_(2)|)=1`
`therefore omega=e^(itheta)` for some `theta, 0 le theta lt 2pi`
Now, `z=(az_(1))/(bz_(2))+(bz_(2))/(az_(1))`
`rArr z=omega+1/omega=e^(itheta)=2costheta`
`rArr` z is purely real i.e., "Im" (z)=0.