Question

If the complex number $z$ satisfies the condition $|z-\frac{12}{z}|=11,$ then the maximum distance from the origin to the point representing $z$ in the Argand plane is

Answer 1

Given : $11=|z-\frac{12}{z}|$
We know that
$|a-b|\ge |a|-|b|$, so
$|z-\frac{12}{z}|\ge |z|-\frac{12}{|z|}\Rightarrow 11\ge |z|-\frac{12}{|z|}\Rightarrow |z{|}^2-11|z|-12\le 0\Rightarrow (|z|+1)(|z|-12)\le 0\Rightarrow -1\le |z|\le 12\Rightarrow 0\le |z|\le 12(\because |z|\ge 0)\Rightarrow |z-(0+0i)|\le 12$

Hence, the maximum distance of $z$ from the origin is $12.$