If `z` is a complex number having least absolute value and `|z-2+2i=1,t h e nz=` `(2-1//sqrt(2))(1-i)` b. `(2-1//sqrt(2))(1+i)` c. `(2+1//sqrt(2))(1-i)""` d. `(2+1//sqrt(2))(1+i)`

Options

A. `(2-1/sqrt(2))(1-i)`
B. `(2-1/sqrt(2))(1+i)`
C. `(2+1/sqrt(2))(1-i)`
D. `(2+1/sqrt(2))(1+i)`

Answer

A

Explanation

We have,
`|z-2+2i|=1`
`rArr |z-(2-2i)|=1`
`rArr` z lies on a circle having center at `(2,-2)` and radius 1.
It is evident from the figure that the required complex number z s given by the point P. We find that OP makes an angle `pi//4` with OX and
OP=OC-CP = `sqrt(2^(2)+2^(2))-1=2sqrt(2)-1`

So, coordinate of P are
`(2sqrt(2)-1)cospi/4, -(2sqrt(2)-1)sinpi/4)`
i.e., `(2-1/sqrt(2)), -(2-1/sqrt(2))`
Hence, `z=(2-1/sqrt(2))+{-(2-1/sqrt(2))}i=(2-1/sqrt(2))1=i`