Question

For all complex numbers $z_1,z_2$ satisfying $\left|z_1\right|=12$ and $|z_2-3-4i|=5$, the minimum value of $\mathbf{|}{\mathit{z}}_{\mathbf{1}}\mathbf{-}{\mathit{z}}_{\mathbf{2}}\mathbf{|}\mathbf{}$is

• A. $4$
• B. $3$
• C. $1$
• D. $2$

Answer 1

The correct option is D

$2$

Finding the minimum value of :

Given, $\left|z_1\right|=12$& $|z_2-3-4i|=5$

Now, we know that $\left|z_1-z_2\right|=\left|z_1\right|-\left|z_2\right|$

Applying this rule,

$\begin{array}{rcl}|z_2-3-4i|& =& 5\\ \left|z_2\right|-\left|-3-4i\right|& =& 5\\ \left|z_2\right|-\sqrt{3^2+4^2}& =& 5\\ \left|z_2\right|-5& =& 5\\ \left|z_2\right|& =& 10\end{array}$

Now,

$\begin{array}{rcl}\left|z_1-z_2\right|& =& \left|z_1\right|-\left|z_2\right|\\ & =& 12-10\\ \left|z_1-z_2\right|& =& 2\end{array}$

Hence, correct option is $\mathbf{\left(}\mathit{D}\mathbf{\right)}$