Question

Let $z$ be a complex number satisfying $|z-5i|\le 1$ such that amp(z) is minimum. Then, $z$ is equal to

• A. $\frac{2\sqrt{6}}{5}+\frac{24i}{5}$
• B. $\frac{24}{5}+\frac{2\sqrt{6i}}{5}$
• C. $\frac{2\sqrt{6}}{5}-\frac{24i}{5}$
• D. None of these

Answer 1

The correct option is A $\frac{2\sqrt{6}}{5}+\frac{24i}{5}$

Let

$z=\frac{2\sqrt{6}}{5}+\frac{24i}{5}$

Then,

$|z-5i|$

$=|\frac{2\sqrt{6}}{5}+\frac{24i}{5}-5i|$

$=|\frac{2\sqrt{6}}{5}+\frac{24i}{5}-5i|$

$=|\frac{2\sqrt{6}}{5}-\frac{i}{5}|$

Amp(z)=

$\sqrt{{\left(\frac{2\sqrt{6}}{5}\right)}^2+{\left(\frac{1}{5}\right)}^2}$

$\sqrt{\frac{24+1}{25}}$

$\sqrt{\frac{25}{25}}$

$=1$