Question

If $|z-1|$ + $|z+3|=8,$ then the range of value of | $z-4|$ is

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

Answer 1

Answer: (C)

We have, $|z-1|+|z+3|=8$
$\Rightarrow |z-(1+0i)|+|z-(-3+0i)|=8$
Thus, if $P,S$ and $S^{\prime }$ are three points in the argand plane representing complex numbers $z,1+0i$ and $-3+0i$, then from Eq. (i), we have
$PS+P{S}^{\prime }=8$
$\Rightarrow P$ lies on the ellipse whose two foci are at $S(1,0)$ and $S^{\prime }(-3,0)$ and major axis $=8$

Now, $PQ=|z-4|$
Clearly, $PQ$ is minimum or maximum according as $P$ coincides with $A(3,0)$ and $A^{\prime \prime }(-5,0)$, respectively.
Thus, $PQ=|z-4|$ varies between $AQ=1$ and $A^{\prime }Q=9$
Hence, $|z-4|\in [1,9]$