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theta in [0,2pi] and z(1), z(2), z(3) ar...

` theta in [0,2pi]` and `z_(1)`, `z_(2)`, `z_(3)` are three complex numbers such that they are collinear and `(1+|sin theta|)z_(1)+(|cos theta|-1)z_(2)-sqrt(2)z_(3)=0`. If at least one of the complex numbers `z_(1)`, `z_(2)`, `z_(3)` is nonzero, then number of possible values of `theta` is

A

Infinite

B

`4`

C

`2`

D

`8`

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` If `z_(1)`, `z_(2)`, `z_(3)` are collinear and `az_(1)+bz_(2)+cz_(3)=0`. Then
`a+b+c=0`
Hence `1+|sintheta|+|costheta|-1-sqrt(2)=0`
`implies |sintheta|+|costheta|=sqrt(2)`
`impliestheta=(pi)/(4)`, `(3pi)/(4)`, `(5pi)/(4)`, `(7pi)/(4)`
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