Minimum distance

$|{ z }_{ 1 }|=12\quad \Rightarrow { z }_{ 1 }\quad lies\quad on\quad a\quad circle\quad with\quad radius\quad 12\quad units\quad and\quad center(0,0)$

$|{ z }_{ 2 }-3-4i|=5\quad \Rightarrow { z }_{ 2 }\quad lies\quad on\quad a\quad circle\quad with\quad radius\quad 5\quad units\quad and\quad center(3,4)$

Therefore minimum value of $\boxed{|{ z }_{ 1 }-{ z }_{ 2 }|=2\quad units}$

This question is based on reverse triangle inequality $| |z_1| - |z_2| | \leq | z_1 - z_2 |$

$| z_2 - ( 3 + 4i ) | = 5$

$| | z_2 | - | 3+ 4i | | \leq | z_2 - ( 3 + 4i ) |$

$| | z_2 | - | 3+ 4i | | \leq 5$

$| |z_2| - 5| \leq 5$

$-5 \leq |z_2| -5 \leq 5$

$-5 -7 \leq | z_2| -5 -7 \leq 5 -7$

$-12 \leq |z_2| - 12 \leq -2$

$-12 \leq |z_2| - |z_1| \leq -2 coz it is given that |z_1| = 12$

$12 \geq |z_1| - |z_2| \geq 2$

$12 \geq | |z_1| - |z_2| | \geq 2$

so the minimum value of $| |z_1| - |z_2| |$ is $2$ .

According to reverse triangle inequality $| |z_1| - |z_2| | \leq | z_1 - z_2 |$

As the minimum value of $| |z_1| - |z_2| |$ is $2$ , the minimum value of $| z_1 - z_2 |$ is $2$ .

distance of Z1 from origin is 12 and distance of 3+4i is from origin is 5..so to minimum the value of distance we take Z1, Z2 and 3+4i is on one st. line..since distance of Z2 from 3+4i is 5.. its distance from Z1 is (12-5)-5=2