Question

If $z$ be a complex number satisfying $\left|\right.z \, -4 \, + \, 8i\left|\right.=4,$ then the least and the greatest value of $\left|\right.z \, + \, 2\left|\right.$ are respectively (where $i=\sqrt{- 1}$ )

• A. $7$ and $16$
• B. $8$ and $17$
• C. $6$ and $14$
• D. $5$ and $13$

Answer 1

Answer: (C)

$\left|\right.z+2\left|\right.=\left|\right.z+2-4+8i+4-8i\left|\right.$
$\left|\right.z+2\left|\right.=\left|\right.z-4+8i+6-8i\left|\right.$
$\left|z + 2\right|\leq \left|\right.z-4+8i\left|\right.+\left|\right.6-8i\left|\right.$
$\leq 4+10=14$
$\left|\right.z+2\left|\right.=\left|\right.z+2-4+8i+4-8i\left|\right.$
$\geq \left|\right.z-4+8i\left|\right.-\left|\right.6-8i\left|\right.$
$\geq \left|4 - 10\right|=6$