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Properties of Modulus

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Question

For all complex numbers $z_{1}$ , $z_{2}$ satisfying $| z_{1} | = 12$ and $| z_{2} - 3 - 4 i | = 5$ , the minimum value of $| z_{1} - z_{2} |$ is:

A

0

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B

2

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C

7

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D

17

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Solution

The correct option is B 2

$C i r c l e 1 : | z_{1} | = 12$
If we consider Circle 1 then center is at (0,0) and radius is 12 units
$C i r c l e 2 : | z_{2} - 3 - 4 i | = 5$
If we consider Circle 2 then center is at (3,4) and radius is 5 units
From the fig.
$m i n | z_{1} - z_{2} | w i l l b e t h e b l a c k c o l o r e d m a r k i n t h e i m a g e i e . R a d i u s o f C i r c l e 1 - D i a m e t e r o f C i r c l e 2$
$m i n | z_{1} - z_{2} | = 12 - 10 = 2$

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