Let Z_1 and Z_2 be two complex numbers satisfying
Question:

Let $Z_{1}$ and $Z_{2}$ be two complex numbers satisfying $\left|Z_{1}\right|=9$ and $\left|Z_{2}-3-4 i\right|=4$. Then the minimum value of $\left|Z_{1}-Z_{2}\right|$ is :

1. 0

2. 1

3. $\sqrt{2}$

4. 2

Correct Option: 1

Solution:

$\left|z_{1}\right|=9, \quad\left|z_{2}-(3+4 i)\right|=4$

$C_{1}(0,0)$ radius $r_{1}=9$

$C_{2}(3,4)$, radius $r_{2}=4$

$\mathrm{C}_{1} \mathrm{C}_{2}=\left|\mathrm{r}_{1}-\mathrm{r}_{2}\right|=5$

$\therefore$ Circle touches internally

$\therefore\left|z_{1}-z_{2}\right|_{\min }=0$