Question

If $|z-25i|\le 15$, then the least positive value of $arg\left(z\right)$ is

• A. $\pi -{\tan }^{-1}\frac{4}{3}$
• B. ${\tan }^{-1}\frac{4}{3}$
• C. $-\pi +{\tan }^{-1}\frac{4}{3}$
  
• D. None of these

Answer 1

Answer: (B)

${\tan }^{-1}\frac{4}{3}$

Since $|z-25i|\le 15,$

therefore, distance between $z$ and $25i$ is less than or equal to $15.$

Thus, point $z$ will lie in the interior and boundary of the circle whose centre is $(0,25)$ and radius is $15.$

Let $OP$ be tangent to the circle at point $P.$

Let $\angle POX=\theta .$ Then, $\angle POC=\theta$

Now, $OC=25,CP=15$

By pythagoras theorem,

$\therefore OP=20.$

Now, $\tan \theta =\frac{OP}{CP}=\frac{20}{15}=\frac{4}{3}$

$\therefore$ Least positive value of arg $z=\theta ={\tan }^{-1}\left(\frac{4}{3}\right)$