Q. Let $\mathcal{z}$ be a complex number satisfying |$\mathcal{\ z -}5i| \leq 1$ such that amp $\mathcal{(z)}$ is minimum. Then $\mathcal{z}$ is equal to
A) $\frac{2\sqrt{6}}{5} + \frac{24i}{5}$
B) $\frac{24}{5} + \frac{2\sqrt{6}i}{5}$
C) $\frac{2\sqrt{6}}{5} - \frac{24i}{5}$
D) None of these
Explanation
(a)
We have, $\mathcal{|z -}5i| \leq 1$
[image:5_image17.emf]
Let $\theta = \angle\ AOX = min.amp\ \left( \mathcal{z} \right),$
$\therefore\ \ \angle AOC = 90{^\circ} - \ \theta$
$\Rightarrow \ \sin{(90{^\circ} - \theta) = \frac{1}{5}}$
$\Rightarrow \ \cos{\theta = \frac{1}{5}}$
$\mathcal{\therefore\ \ z =}OA\cos{\theta +}i\ OA\sin\theta$
$\mathcal{\Rightarrow \ \ z =}\sqrt{5^{2} - 1}\left( \frac{1}{5} \right) + i\ \sqrt{5^{2} - 1}\sqrt{1 - \frac{1}{5^{2}}}$
$= \frac{2\sqrt{6}}{5}(1 + i\ 2\sqrt{6})$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.