Question

If |Z-2+1| ≤2, the greatest and least values of |z + 4| are

• A. √41+4, √(41 )-4
• B. √37+7, √37 -7
• C. √37+2, √37-2
• D. √41+5, √41-5

Answer 1

The correct option is C

√37+2, √37-2

Given |z-2+i| ≤ 2

⟹ |z+4-6+i| ≤2

|(z+4)- (6-i)| ≤2

Using |(|$Z_1$|-|$Z_2$|)| ≤ |$Z_1$-$Z_2$|

⟹ |(|z+4|- |6-i|)| ≤ |(z+4)- (6-i)| ≤2

⟹ |(|z+4|-|6-i|)|≤2

-2 ≤ |z+4| - $\sqrt{37}$ ≤ 2

$\sqrt{37}$ - 2 ≤ |z+4| ≤ $\sqrt{37}$ +2

so, the greatest value of |z+4| = $\sqrt{37}$ - 2