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 $|z_1-z_2|$ is

• A. $0$
• B. $2$
• C. $7$
• D. $17$

Answer 1

Answer: (B)

$2$

$C_1(0,0)$ is center of bigger circle and $C_2(3,4)$ is center of smaller circle

$C_{1}B=r_1=12$ (radius of bigger circle)

$C_{2}A=r_2=5$ (radius of smaller circle)

$C_1{C}_2=5,$ minimum value of $|z_1-z_2|=AB$

now, $C_1{C}_2+C_{2}A+AB=C_{1}B\Rightarrow AB=12-5-5=2$