Complex Numbers
Argument
MMTS_Full_Test_06
Grade 12
Question:
Let $z_1$ and $z_2$ be two complex numbers such that $|z_1|=1$ and $|z_2|=10$. If $\theta=\arg\left(\dfrac{z_1-z_2}{z_2}\right)$, then maximum value of $\tan^2\theta$ is
$\dfrac{1}{10}$
$\dfrac{1}{100}$
$\dfrac{1}{99}$
$\dfrac{10}{99}$
Step-by-Step Solution
Key Concept: $\frac{z_1-z_2}{z_2}=\frac{z_1}{z_2}-1$; let $w=z_1/z_2$, $|w|=1/10$
$w=z_1/z_2$, $|w|=1/10$. $w-1$ is a complex number; arg of $w-1$ where $w$ lies on circle of radius $1/10$ centered at origin. Max $\tan\theta=\frac{1/10}{\sqrt{1-1/100}}=\frac{1}{\sqrt{99}}$. $\tan^2\theta=1/99$.
Correct Answer: 3