Question

Let $z _1$ and $z _2$ be two complex numbers such that $\left| z _1\right|=1$ and $\left| z _2\right|=10$. If $\theta=\arg \left(\frac{ z _1- z _2}{ z _2}\right)$ then maximum value of $\tan ^2 \theta$ can be expressed as $\frac{m}{n}$ (where $m$ and $n$ are coprime), find the value of $(100 m-n)$.

Answer 1

Put $\frac{ z _1}{ z _2}= z$
$\therefore\left|\frac{z_1}{z_2}\right|=|z| \Rightarrow|z|=\frac{1}{10}$
Also, $\theta=\arg \left(\frac{ z _1}{ z _2}-1\right)=\arg ( z -1)$
$\left.\therefore \tan ^2 \theta\right]_{\max }=\frac{1 / 100}{99 / 100}=\frac{1}{99} \equiv \frac{m}{n} \text { (Given). }$