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Distance Formula Using Complex Numbers

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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

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Solution

The correct option is C $2$
Ans. $(b)$ . The two circles are $C_{1} (0, 0)$ , $r_{1} = 12$
$C_{2} (3, 4)$ , $r_{2} = 5$
and it passes through origin, the center of $C_{1}$ .
$C_{1} C_{2} = 5 < r_{1} - r_{2} = 7$
Hence circle $C_{2}$ lies inside circle $C_{1}$ .
Therefore minimum distance between them is
$A B = C_{1} B - C_{1} A - r_{1} - 2 r_{2} = 12 - 10 = 2$

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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

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