If z = x + iy and x² + y² = 16, then the range of ||x| - |y|| is

Here, $\mathrm{x}=4\cos \mathrm{\theta }, \mathrm{y}=4\sin \mathrm{\theta }$

$\begin{array}{c}\therefore \parallel \mathrm{x}|-|\mathrm{y}\left|\right|=\left|4\right|\cos \mathrm{\theta }|-4|\sin \mathrm{\theta }\parallel \\ =4\parallel \cos \mathrm{\theta }|-|\sin \mathrm{\theta }\parallel \\ =4\sqrt{1-2|\cos \mathrm{\theta }\parallel \sin \mathrm{\theta }|}\\ =4\sqrt{1-|\sin 2\mathrm{\theta }|}\end{array}$

Hence, the range is [0, 4].