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Definition of Relations

If z1 ,z2 a...

Question

If $z_{1}, z_{2} a n d z_{3}$ be three complex numbers such that $| z_{1} + 1 | \leq 1, | z_{2} + 2 | \leq 2 a n d | z_{3} + 4 | \leq 4$ , then the maximum value of $| z_{1} | + | z_{2} | + | z_{3} |$ is

A

7

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B

10

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C

12

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D

14

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Solution

The correct option is D $14$
Property: $| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} |$

$| (z_{1} + 1) - 1 | \leq | z_{1} + 1 | + | - 1 |$
$| z_{1} | \leq | z_{1} + 1 | + 1$
$| z_{1} | \leq 1 + 1$
$| z_{1} | \leq 2$ ............ (1)

$| (z_{2} + 2) - 2 | \leq | z_{2} + 2 | + | - 2 |$
$| z_{2} | \leq | z_{2} + 2 | + 2$
$| z_{2} | \leq 2 + 2$
$| z_{2} | \leq 4$ ............ (2)

$| (z_{3} + 4) - 4 | \leq | z_{3} + 4 | + | - 4 |$
$| z_{3} | \leq | z_{3} + 4 | + 4$
$| z_{3} | \leq 4 + 4$
$| z_{3} | \leq 8$ ............ (3)

From equation (1), (2) and (3),
$| z_{1} | + | z_{2} | + | z_{3} | \leq 2 + 4 + 8$
$| z_{1} | + | z_{2} | + | z_{3} | \leq 14$

So the maximum value of $| z_{1} | + | z_{2} | + | z_{3} |$ is $14$

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