Question

lf $|z-1|+|z+3|\le 8$, then the range of values of $|z-4|$ is

• A. $(0,8)$
• B. $[0,8]$
• C. $[1,9]$
• D. $[5,9]$

Answer 1

The correct option is C $[1,9]$

Given,

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

$Forz<-3:$

$-(z-1)-(z+3)\le 8$

$-5\le z<-3$

$For-3\le z<1$

$-(z-1)+z+3\le 8$

$-3\le z<1$

$Forz\ge 1$

$z-1+z+3\le 8$

$1\le z\le 3$

$\therefore -5\le z<-3or-3\le z<1or1\le z\le 3$

$\Rightarrow -5\le z\le 3$

For $|z-4|$

interval is $4-3\le z\le 4-(-5)$

$\therefore [1,9]$ is the range