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Let A,B,C be three sets of complex numbe...

Let A,B,C be three sets of complex number as defined below
`A={z:lm (z) ge 1}`
`B={z:|z-2-i|=3}`
`C={z:Re((1-i)z)=sqrt(2)}`
Let z be any point in `A cap B cap C` and let w be any point satisfying `|w-2-i|lt 3` Then `|z|-|w|+3` lies between

A

`-6 and 3`

B

`-3 and 6`

C

`-6 and 6`

D

`-3 and 9 `

Text Solution

Verified by Experts

The correct Answer is:
D
Since `|w-(2+i)| lt 3 rArr |w|-|2+i|lt 3 `
`rArr -3+sqrt(5)lt |w| lt 3 + sqrt(5)` ltbr `rArr -3-sqrt(5)lt -|w|lt 3 -sqrt(5)`
Also `|z-2(2+i)|=3`
`rArr -3 + sqrt(5) le |z| le 3 +sqrt(5)`
`therefore -3 lt |z| -|w|+3 lt 9 `
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