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Equation of Circle in Complex Form

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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 B
2

Given : $| z_{1} | = 12.$ It is a circle with centre at $O (0, 0)$ and radius, $O B = 12$

Also, $| z_{2} - 3 - 4 i | = 5$ , so, it is a circle with centre at $C (3, 4)$ and radius $= 5$

$\Rightarrow O D = 2 \times 5 = 10$

$| z_{1} - z_{2} |_{m i n} = B D = O B - O D$

$= 12 - 10$

$= 2$

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