Question

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

• A. $0$
• B. $1$
• C. $3$
• D. $5$

Answer 1

The correct option is D

$5$

The explanation for the correct option:

It is given that, $\left|z-3-2i\right|\le 2$.

Simplify the given expression.

$$\begin{gathered} \left|2z-6+5i\right|=2\left|z-3+\frac{5}{2}i\right| \\ \Rightarrow \left|2z-6+5i\right|=2\left|z-3-2i+\frac{9}{2}i\right| \end{gathered}$$

It is known that $\left|\left|z_1\right|-\left|z_2\right|\right|\le \left|z_1+z_2\right|$.

Thus $2\left|z-3-2i+\frac{9}{2}i\right|\ge 2\left|\left|z-3-2i\right|-\left|\frac{9}{2}i\right|\right|$.

$$\begin{gathered} \Rightarrow \left|2z-6+5i\right|\ge 2\left|2-\sqrt{{\left(\frac{9}{2}\right)}^2}\right|\left[\because \left|z-3-2i\right|\le 2\right] \\ \Rightarrow \left|2z-6+5i\right|\ge 2\left|2-\frac{9}{2}\right| \\ \Rightarrow \left|2z-6+5i\right|\ge 2\left|\frac{4-9}{2}\right| \\ \Rightarrow \left|2z-6+5i\right|\ge 2\left|\frac{-5}{2}\right| \\ \Rightarrow \left|2z-6+5i\right|\ge 2\times \frac{5}{2} \\ \Rightarrow \left|2z-6+5i\right|\ge 5 \end{gathered}$$

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

Hence, (D) is the correct option.