Find the range of values of $|z|$ and $\arg (z)$ for $$|z-4-4i| = 2 \sqrt{2}.$$
I'm aware that you can solve this geometrically by drawing a circle on the argand diagram and finding out the information from there. I was curious if there is an algebraic method for solving the problem of finding the range of values for the modulus and argument (i.e. without needing to consider it in the argand plane)?
Thanks
Best Answer
Hint. Note that the equation is solved by $$z=4+4i+2 \sqrt{2}(\cos(t)+i\sin(t))=4+2 \sqrt{2}\cos(t) +i\left(4+2 \sqrt{2}\sin(t)\right)$$ with $t\in [0,2\pi)$. Hence $$|z|^2=(4+2 \sqrt{2}\cos(t))^2 +(4+2 \sqrt{2}\sin(t))^2=40+16\sqrt{2}(\cos(t)+\sin(t))\\ =40+32\sin(t+\pi/4).$$ Hence $$40-32\leq |z|^2\leq 40+32.$$ Can you take it from here?
As regards the argument consider the ratio $$\tan(\arg(z))=\frac{\mbox{Im}(z)}{\mbox{Re}(z)}=\frac{4+2 \sqrt{2}\sin(t)}{4+2 \sqrt{2}\cos(t)}.$$ Finally you should find that the minimum argument is $\pi/12$. What is the maximum argument?