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/z`
B. `barz`
C. `1/barz`
D. none of these

Answer

C

Explanation

Let the affix of A be `z_(1)`. Then,
`OA.OB=1 rArr |z||z_(1)| =1 rArr |z_(1)|=1/|z|`
Clearly, arg`(z_(1))="arg"(z)=0` (say)
`therefore z_(1)=1/|z|(cos+isintheta)` `[therefore |z_(1)|=1/|z|]`
`rArr z_(1) = 1/|z|^(2)(|z|(cos+isintheta)}`
`rArr z_(1)=1/|z|^(2)z=z/(zbarz)=1/z`