1.

Let z and omega be two complex numberssuch that |z|le 1, |omega| le 1 and |z+ iomega| = |z_(1)-z_(2)|is equal to

Answer»

`(2)/(3)`
`(sqrt(5))/(3)`
`(3)/(2)`
`(2sqrt(5))/(3)`

Solution :We have `2=|z+iomega|le|z|+|omega|""(THEREFORE |z_(1)+z_(2)|le|z_(1)|+|z_(2)|)`
`therefore |z|+|omega|le2""...(1)`
But it is given that `|z|le1 and |omega|le1`.
`rArr |z|+|omega|le2""...(2)`
From (1) and (2),
`|z|=|omega|=1`
Also, `|z+iomega|=|z-(ibaromega)|`
`rArr |z-(-iomega)|=|z-(ibaromega)|`
This means that z lies on perpendicularbisector of theline segment JOINING `(-iomega) and (ibaromega)`, which is real axis, as `(-iomega) and (ibaromega)` are CONJUGATE to each other.
For`z, Im(z) = 0`
If `z =x, " then " |z|le1`
`rArr x^(2)le1`
`rArr -1le x le 1`
Therefore, (3) is the correct option.


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