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 $(100m-n)$.

Answer 1

Answer: 1

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