Question

The greatest and least values of $|z+1|$ if $|z+4|\le 3$ are 'a' and 'b' , Find $a-b$

Answer 1

$|z+1|=|z+4-3|=|(z+4)+(-3)|$ $\le |z+4|+|-3|=|z+4|+3$ $\le 3+3=6$ $[\because |z+4|\le 3]$
Hence the greatest value of |z+1| is 6
$|z+1|=|z+4-3|\ge |z+4|-\left|3\right|=3-3=0.$
Hence least value is zero.
Therefore, $a-b=6$
Ans: 6