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The value of k for which the inequality ...

The value of k for which the inequality `| Re (z) | + | Im (z)| leq lambda |z|` is true for all `z in C` , is

A

2

B

`sqrt(2)`

C

1

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let `z=r(costheta+isintheta)`. Then, `|z|=r`and arg(z) `=theta`
Now, `|"Re"(z)|+|"Im"(z)| le |rcostheta|+|rsintheta|`
`rArr |"Re"(z)|+|"Im"(z)|=r|costheta|+|sintheta|`
`rArr |"Re"(z)|+|"Im"(z)|^(2)=r^(2)|1+|sin2theta|`
`rArr |"Re"(z)|+|"Im"(z)|^(2) le sqrt(2)r`
`rArr |"Re"(z)|+|"Im"(z)| le sqrt(2)r`
`rArr |"Re"(z)|+|"Im"(z)| le sqrt(2)|z|`
Hence, `lambda=sqrt(2)`.
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