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Let Z(1) and Z(2) be two complex numbers...

Let `Z_(1)` and `Z_(2)` be two complex numbers satisfying `|Z_(1)|=9` and `|Z_(2)-3-4i|=4`. Then the minimum value of `|Z_(1)-Z_(2)|` is

A

1

B

2

C

`sqrt(2)`

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
Clearly `|z_1|=9 ` represents a circle having centre `C_1(0,0)` and radius `r_1=9`
and `|z_2-3-4i|=4` represents a circle having centre `C_2(3,4)` and radius `r_2=4`
The minimum value of f`|z_1-z_2| ` is equals to minimum distance between circless `|z_1|=9` and `|z_2-3-4i|=4`
`therefore C_1 C_2= sqrt((3-0)^2(4-0)^2)= sqrt(25)=5`
and `|r_1-r_2|=|9-4|=5 rArr C_1C_2 = |r_1-r_2|`
`therefore `Circles touches each other internally .
Hence , `|z_1-z_2|_(min)=0`
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