Question

If $\left|z\right|=5$, then the points representing the complex number $-i+\frac{15}{z}$ lies on the circle

• A. whose center is $(0,1)$ and radius=$3$
  
• B. whose center is $(0,-1)$ and radius=$3$
• C. whose center is $(1,0)$ and radius=$15$
• D. whose center is $(-1,0)$ and radius=$15$

Answer 1

The correct option is B whose center is $(0,-1)$ and radius=$3$

let $z=x+iy$

$\Rightarrow -i+\frac{15}{z}=-i+15\frac{\overline{z}}{|z{|}^2}$

$\Rightarrow -i+\frac{15\overline{z}}{25}$

$\Rightarrow -i+\frac{3\overline{z}}{5}$

$\Rightarrow -i+\frac{3}{5}(x-iy)$

$\Rightarrow \frac{3}{5}\left(z\right)-i$

This point lies on a circle wth cennter $(0,-1)$ and radius $3$