Question

For all complex numbers $z_1,z_2$ satisfying $\left|z_1\right|=12$ and $|z_2-3-4i|=5$, then minimum value of $|z_1-z_2|$ is

• A. $0$
• B. $2$
• C. $7$
• D. $17$

Answer 1

The correct option is B $2$

Given:-

$z_1$ and $z_2$ are the complex number.

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

To find minimum value of $|z_1-z_2|$

Solution:-

We know that

$|z_1-z_2|\ge \left|z_1\right|-\left|z_2\right|$

and

$|z_2-3-4i|\ge \left|z_2\right|-|3+4i|$

$|z_2-3-4i|\ge \left|z_2\right|-\sqrt{{\left(3\right)}^2+{\left(4\right)}^2}\left\{|a+bi|\right\}=\sqrt{a^2+b^2}5\ge \left|z_2\right|-5\left|z_2\right|\ge 10$

for minimum value.$z_2=10$

$\therefore$minimum value of $|z_1-z_2|\ge \left|z_1\right|-\left|z_2\right||z_1-z_2|\ge 12-10|z_1-z_2|\ge 2$

Hence the minimum value of $\left|z_1\right|-\left|z_2\right|=2$