Complex Numbers 2 Question 47

48.

If $z$ is any complex number satisfying $|z-3-2 i| \leq 2$, then the maximum value of $|2 z-6+5 i|$ is

(2011)

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Answer:

Correct Answer: 48. (5)

Solution:

  1. Given, $|z-3-2 i| \leq 2$

To find minimum of $|2 z-6+5 i|$

or $2\left|z-3+\frac{5}{2} i\right|$, using triangle inequality

i.e. ||$z _1|-| z _2|| \leq\left|z _1+z _2\right|$

$\therefore \quad\left|z-3+\frac{5}{2} i\right|=\left|z-3-2 i+2 i+\frac{5}{2} i\right|$

$=\left|(z-3-2 i)+\frac{9}{2} i\right|$

$\geq|z-3-2 i|-\frac{9}{2}|\geq| 2-\frac{9}{2} \mid \geq \frac{5}{2}$

$\Rightarrow \quad\left|z-3+\frac{5}{2} i\right| \geq \frac{5}{2}$ or $|2 z-6+5 i| \geq 5$



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