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Properties of Modulus

If | z 1|=2,|...

Question

If $| z_{1} | = 2, | z_{2} | = 3,$ then the maximum value of $| z_{1} + z_{2} + 5 + 12 i |$ is

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Solution

Given $| z_{1} | = 2, | z_{2} | = 3$
We know, $| z_{1} + z_{2} + \dots + z_{n} | \leq | z_{1} + | z_{2} | + \dots + | z_{n} |$
$∴ | z_{1} + z_{2} + (5 + 12 i) | \leq | z_{1} + | z_{2} | + | 5 + 12 i |$
$\Rightarrow | z_{1} + z_{2} + (5 + 12 i) | \leq 2 + 3 + 13$
$\Rightarrow | z_{1} + z_{2} + (5 + 12 i) | \leq 18$

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