$| z - 3 | < 1$ and $| z - 4 i | > M$ then find the positive real value of $M$ for which these exist at least one complex number $z$ satisfy both the equation.

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Solution

$| z - 3 | < 3$ is the interior area of the circle having centre at $(3, 0)$ and radius $1$ .
Similarly, $| z - 4 i | > M$ is the exterior area of the circle of radius $M$ and having centre at $(0, 4)$ .

Distance between $(3, 0)$ and $(0, 4) = 5$

$∴$ From the figure the value of $M$ is $= 5 - 1 = 4$
$M = 4$ is the value for $M$ for which there exists at least one complex number $z$ satisfies both equations.

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