Question

Consider the complex numbers satisfying $|z-25i|\le 15$ then

• A. Complex number with least argument has argument ${tan}^{-1}\left(\frac{4}{3}\right)$
• B. Complex number with maximum argument has amplitude $\pi -{\tan }^{-1}\left(\frac{4}{3}\right)$
• C. Complex number least magnitude is 10
• D. Complex number maximum magnitude is 40

Answer 1

Answer: (A), (B)

The correct options are
A Complex number with least argument has argument ${tan}^{-1}\left(\frac{4}{3}\right)$
B Complex number with maximum argument has amplitude $\pi -{\tan }^{-1}\left(\frac{4}{3}\right)$

Given,

$|z-25i|\le 15$

$||z-z_1|\le r$

$z_1\to$ centre of circle

$r\to$ radius

$z_1\to 25i$

centre $=(0,25)$

$r=15$

Refer fig, 1 and 2.

${\theta }_1=90-\theta$

$=9-37$

$={53}^0$

$minarg\left(z\right)={\theta }_1={53}^0$

$|minarg\left(z\right)|={\tan }^{-1}\left(\frac{4}{3}\right)$

${\theta }_2=2\theta -{\theta }_1$

$={53}^0+2\left({37}^0\right)$

$={127}^0$

$minarg\left(z\right)={127}^0$

$|minarg\left(z\right)|=\pi -{\tan }^{-1}\left(\frac{4}{3}\right)$

Refer fig, 3

$\sin \theta =\frac{r}{25}$

$\sin \theta =\frac{15}{25}$

$\sin \theta =\frac{3}{5}$

$\theta ={37}^0$

$\cos \theta =\frac{4}{5}$

$\tan \theta =\frac{3}{4}$

$\theta ={\tan }^{-1}\left(\frac{4}{3}\right)$