Let `z_(1)and z_(2)`be two complex numbers represented by points on circles `|z|=1and |z|=2` respectively, then

Options

A. max `|2z_(1)+z_(2)|=4`
B. min `|z_(1)-z_(2)|=1`
C. `|z_(2)+1/z_(1)| le 3`
D. all of the above.

Answer

D

Explanation

We have, `|z_(1)|=1`and `|z_(2)|=2.`
Now, `|2z_(1)+z_(2)| le |2z_(1)|+|z_(2)|`
`rArr |2z_(1)+z_(2)| le 2 xx 1 +2`
`rArr |2z_(1) +z_(2)| le 4`
Thus, option (a) is true.

Suppose `z_(1)` and `z_(2)` are represented by points P and Q respectively. Then,
PQ=`|z_(1)-z_(2)|`
Clearly, PQ will be minimum,If O,P and Q are in a straight line and in that case
`PQ=OQ-OP=2-1=1 rArr |z_(1)-z_(2)|=1`
Hence, min`|z_(1)-z_(2)|`
Clearly, PQ will be minimum, If O,P and Q are in a straight line and in that case.
`PQ=OQ-OP=2-1 =rArr |z_(1)-z_(2)|=1`
Hence, min `|z_(1)-z_(2)|=1`
Thus, option (b) is correct.
Now, `|z_(2)+1/z_(1)| le |z_(2)|+1/|z_(1)|=2+1=3` `[therefore |z_(1)|=1, |z_(2)|=2]`
So, option c is also correct.