If - z -2- i -5- then the maximum value of -3 z -9-7 i - isA- 15B- 20C- 5D- 10

Given, |z+2 i|=5 …(i) Now, |3z+9 7i|=|3z+6 3i+3 4i| We know that, |z_1+z_2| ≤ |z_1|+|z_2| ∴ |3z+6 3i+3 4i| ≤ |3z+6 3i|+|3 4i| ⇒ |3z+9 7i| ≤ 3|z+2 i| +|3 4i| ⇒ ...

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Standard XII

Mathematics

Properties of Modulus

If | z +2- i ...

Question

If $| z + 2 - i | = 5,$ then the maximum value of $| 3 z + 9 - 7 i |$ is

10

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20

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Solution

The correct option is C $20$
Given,
$| z + 2 - i | = 5 \dots (i)$

Now, $| 3 z + 9 - 7 i | = | 3 z + 6 - 3 i + 3 - 4 i |$

We know that,
$| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} |$
$∴ | 3 z + 6 - 3 i + 3 - 4 i | \leq | 3 z + 6 - 3 i | + | 3 - 4 i | \Rightarrow | 3 z + 9 - 7 i | \leq 3 | z + 2 - i | + | 3 - 4 i |$
$\Rightarrow | 3 z + 9 - 7 i | \leq 3 (5) + 5$ [from $(i)$ ]
$\Rightarrow | 3 z + 9 - 7 i | \leq 20$

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Properties of Modulus

Standard XII Mathematics

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If |z+2−i|=5, then the maximum value of |3z+9−7i| is

The correct option is C 20Given, |z+2−i|=5…(i) Now, |3z+9−7i|=|3z+6−3i+3−4i| We know that, |z1+z2|≤|z1|+|z2| ∴|3z+6−3i+3−4i|≤|3z+6−3i|+|3−4i|⇒|3z+9−7i|≤3|z+2−i|+|3−4i| ⇒|3z+9−7i|≤3(5)+5 [from (i)] ⇒|3z+9−7i|≤20

If $| z + 2 - i | = 5,$ then the maximum value of $| 3 z + 9 - 7 i |$ is