For all complex numbers `z_1,z_2` satisfying `|z_1|=12` and `|z_2-3-4i|=5`, find the minimum value of `|z_1-z_2|`

Answer

2

Explanation

`|z_(1)|=12rArr z_(1)`lies on a circle with centre (0,0) and radius 12 units, and `|z_(2)-3-4i|=5rArr z_(2)` lies on a circle with centre (3,4) and radius 5 units.
From the figure it is clear that `|z_(1)-z_(2)|`, that is , distance between `z_(1)` and `z_(2)` will be minimum when they lie at A and B, respectively, i.e., on diagram as shown. Then
`|z_(1)-z_(2)|=AB=OA=12-2(5)=2`