Let `O`, `A`, `B` be three collinear points such that `OA.OB=1`. If `O` and `B` represent the complex numbers `O` and `z`, then `A` represents

Options

A. `(1)/(barz)`
B. `(1)/(z)`
C. `barz`
D. `z^(2)`

Answer

A

Explanation

`(a)` Let `A` represents `z_(1)`.
Since `OA.OB=1 :. |z_(1)-0|.|z-0|=1`
`implies |z_(2)|=(1)/(|z|)`
Also, `arg((z_(1)-0)/(z-0))=0impliesarg((z_(1))/(z))=0`
`impliesargz_(1)=argz`
If `theta` is the argument of `z`, then
`z=|z|e^(itheta)`
`:. z_(1)=(1)/(|z|)e^(etheta)=(1)/(|z|^(2))|z|e^(itheta)-(z)/(zbarz)=(1)/(barz)`
`:. A` is `(1)/(baraz)`