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 D 2
The two circles whose centre and radius are $C_{1} (0, 0), r_{1} = 12, C_{2} (3, 4), r_{2} = 5$ and it passes through origin ie, the centre of $C_{1}$ .
Now, $C_{1} C_{2} = \sqrt{3^{2} + 4^{2}} = 5$
and $r_{1} - r_{2} = 125 = 7$
$∴ C_{1} C_{2} < r_{1} - r_{2}$
Hence, circle $C_{2}$ lies inside the circle $C_{1}$ .
From figure 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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