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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Solution:
- 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 $|2 z-6+5 i| \geq 5$
