Complex Number Help (1 Viewer)

cineti970128 · Dec 4, 2012

Find the greatest value of |z| if the complex number satisfies |z-4/z| = 8
i did it like

|z²- 4| = 8|z|
Using triangular inequality
8|z| < or = |z|²- 4

Hence when equlity holds maximum value
|z|²- 8|z|-4 = 0
Hence solving the equation which gives maximum |z|?

I am not sure if I am wrong could you tell me your working and your answer ASAP

Carrotsticks · Dec 4, 2012

We use the Reverse Triangle Inequality:

$\\ |z^2-4| \geq |z|^2 - 4 \\\\ 8|z| \geq |z|^2 - 4 \\\\ |z|^2 - 8 |z| - 4 \leq 0 \\\\ $Solving this quadratic for the bounds, we have: $ \\\\ |z| = \frac{8 \pm \sqrt{80}}{2} = 4 \pm 2 \sqrt{5} \\\\ $Hence, solving the inequality, we have: $ \\\\ 4 - 2 \sqrt{5} \leq |z| \leq 4 + 2 \sqrt{5} \\\\ $However, we cannot have $ 4 - 2 \sqrt{5} $ since that value is less than 0, so our lower bound is just 0 and hence $ 0 \leq |z| \leq 4+2\sqrt{5} \\\\ $From here, we observe that the maximum value of $ |z| $ is $ |z| = 4 + 2\sqrt{5}$

cineti970128 · Dec 4, 2012

yeah eventually i got the same answer
Cheers Carrotsticks ^^

Complex Number Help (1 Viewer)

cineti970128

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Carrotsticks

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