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Geometrical Representation of a Complex Number

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Question

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

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Solution

The correct option is B $2$
The two circles are $C_{1} (0, 0), r_{1} = 12; C_{2} (3, 4), r_{2} = 5$ and it passes through origin.
The distance between centres is $C_{1} C_{2} = \sqrt{3^{2} + 4^{2}} < 7 = r_{1} - r_{2}$
Hence, circle $C_{2}$ lies inside circle $C_{1}$ .
Therefore, the 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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