Question

If $\left|z-1\right|+\left|z+3\right|\le 8$ then the range of values of $\left|z-4\right|$

• A. $[1,7]$
• B. $[1,8]$
• C. $[1,9]$
• D. $[2,5]$

Answer 1

The correct option is A $[1,7]$

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

Using the triangle inequality

$|z_1\pm z_2|\le |z_1|+|z_2|$

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

$⟹|z-1+z+3|\le 8$

$⟹|z+1|\le 4$

Using triangle inequality again

$|z|+1\le 4⟹|z|\le 3$

So, the maximum value of $|z_1{+}_2|$ is $|z_1|+|z_2|$

And the minimum value of $|z_1{+}_2|$ is $|z_1|-|z_2|$

Hence the maximum value of $|z-4|$ is $|z|+|4|=3+4=7$

the minimum value of $|z-4|$ is $|z|-|4|=3-4=-1$

Hence the range of $|z-4|$

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