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Let A,B and C be three sets of complex n...

Let A,B and C be three sets of complex numbers as defined below: `{:(,A={z:Im(z) ge 1}),(,B={z:abs(z-2-i)=3}),(,C={z:Re(1-i)z)=3sqrt(2)"where" i=sqrt(-1)):} `
The number of elements in the set `A cap B cap C`, is

A

0

B

1

C

2

D

`infty`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `z=x+iy`. Then,
`(1-i)z=(1-i)(x+iy)=(x+iy)+i(y-x)`
`rArr "Re"{(1-i)z}=x+y`
`therefore {(1-i)z}=sqrt(2) rArr x+y=sqrt(2)`

Clearly, A is the set of all points lying on or above the line y=1 in xy-plane, B is the set of all points lying on the circle having center at (2,1) and radius 3 and C is the set of all points on the line `x+y=sqrt(2)`.
The line `x+y=sqrt(2)` intersects the circle `|z-(2+i)|=3` at P and Q out of which P lies in the region represented by Im `(z) ge 1`.
`therefore A frown B frown C` contains exactly one point.
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