Question

If $z$ is any complex number satisfying $\left|z-3-2i\right|\le 2$, then the minimum value of $\left|2z-6+5i\right|$ is

• A. $2$
• B. $1$
• C. $3$
• D. $5$

Answer 1

The correct option is D

$5$

Explanation for the correct option.

Find the minimum value of $\left|2z-6+5i\right|$.

The expression $\left|2z-6+5i\right|$ can be written as

$\begin{array}{rcl}\left|2z-6+5i\right|& =& 2\left|z-3+\frac{5}{2}i\right|\\ & =& 2\left|z-3-2i+2i+\frac{5}{2}i\right|\\ & =& 2\left|\left(z-3-2i\right)+\frac{9}{2}i\right|\end{array}$

Now, for two complex number $z_1$ and $z_2$ the triangle inequality can be given as: $\left|\left|z_1\right|-\left|z_2\right|\right|\le \left|z_1+z_2\right|$.

So the minimum value of $\begin{array}{rcl}& & 2\left|\left(z-3-2i\right)+\frac{9}{2}i\right|\end{array}$ is given as:

$\begin{array}{rcl}2\left|\left(z-3-2i\right)+\frac{9}{2}i\right|& =& 2\left|\left|\left(z-3-2i\right)\right|-\left|\frac{9}{2}i\right|\right|\\ & =& 2\left|2-\frac{9}{2}\right|\left[\left|z-3-2i\right|\mathbf{\le }\mathbf{2}\right]\\ & =& 2\left|-\frac{5}{2}\right|\\ & =& 2\times \frac{5}{2}\\ & =& 5\end{array}$

So, the minimum value of $\left|2z-6+5i\right|$ is $5$.

Hence, the correct option is D.