Question

If $|z|\le 4$ then find the maximum value of $|iz+3-4i|$

Answer 1

Taking max. $\left|z\right|=4$

$\to |iz+3-4i|=\left|i\right||z+\frac{3}{i}-4|$

$=|z-3i-4|=|\left(z\right)+(-3i-4)|$

Using triangle inequality, we get

$|\left(z\right)+(-3i-4)|\le \left|z\right|+|-3i-4|$

$|iz+3-4i|\le 4+5$

$|iz+3-4i|\le 9$

Therefore, max. value is $9$