Question

If $|z-1|+|z-3|\le 8$, then find the range of $|z-4|$.

Answer 1

Given :$|z+1|+|z-3|\le 8$

To find : Range of $|z-4|$

Triangle Inequality : $|z_1+z_2|\le |z_1|+|z_2|$

$\to |z+1+2-3|\le 8$

$\to |2z-2|\le 8$

$\to |z-1|\le 4$

$\to |z|\le 3$

Now,

$\to ||z|-4|\le |z-4|\le ||z|+4|$

$\to 1\le |z-4|\le 7$

$\therefore |z-4|\in [1,7]$