Question

Locate the complex number $z=x+iy$ for which $\frac{1}{2}<lo{g}_3|z-5i|<1$

• A. The region included between two concentric circles having centre at 0 + 5i and radii $\sqrt{2}$ and 2 units respectively.
• B. The region included between two concentric circles having centre at 0 - 5i and radii $\sqrt{3}$ and 3 units respectively.
• C. The region included between two concentric circles having centre at 0 - 5i and radii $\sqrt{2}$ and 2 units respectively.
• D. The region included between two concentric circles having centre at 0 + 5i and radii $\sqrt{3}$ and 3 units respectively.

Answer 1

The correct option is D The region included between two concentric circles having centre at 0 + 5i and radii $\sqrt{3}$ and 3 units respectively.

Taking antilog we get
$\sqrt{3}<|z-5i|<3$
Now
$|z-5i|>\sqrt{3}$ depicts all the points outside the circle centered at $(0,5)$ and having a radius of $\sqrt{3}$
Similarly
$|z-5i|<3$ depicts all the points inside the circle centered at $(0,5)$ and having a radius of $3$.
Hence the inequality
$\sqrt{3}<|z-5i|<3$.
Depicts all the points between the above two co-centric circles.