Question

If the complex number z satisfies the condition z>=3 then the least value of

Answer 1

Answer:

We have,

∣∣∣z+1z∣∣∣≥∣∣∣|z|−∣∣1z∣∣∣∣∣|z+1z|≥||z|−|1z||

[∵|z1+z2|≥∣∣|z1|−|z2|∣∣.][∵|z1+z2|≥||z1|−|z2||.]

⟹∣∣∣z+1z∣∣∣≥∣∣∣|z|−1|z|∣∣∣...(i)⟹|z+1z|≥||z|−1|z||...(i)

As, |z|≥3|z|≥3

⟹1|z|≤13⟹1|z|≤13

⟹−1|z|≥−13⟹−1|z|≥−13

∴|z|−1|z|≥3−13=83∴|z|−1|z|≥3−13=83

⟹∣∣∣|z|−1|z|∣∣∣≥∣∣83∣∣=83...(ii)⟹||z|−1|z||≥|83|=83...(ii)

Thus, from (i) & (ii), we can conclude that,

∣∣∣z+1z∣∣∣≥83.†|z+1z|≥83.†

Hence, the required minimum value is,

=83.=83.

Hope, you'll understand..!!