Question

Let z be a complex number find the value of $\left|\frac{z-i}{z+2i}\right|=1$ and$\left|z\right|=\frac{5}{2}$. Then the value of $\left|z+3i\right|$ is :

• A. $\sqrt{10}$
• B. $\frac{7}{2}$
• C. $\frac{15}{4}$
• D. $\frac{2}{\sqrt{3}}$

Answer 1

The correct option is B

$\frac{7}{2}$

Explanation for the correct option:

Calculating the value of $\left|z+3i\right|$:

Given

$\begin{array}{rcl}\left|\frac{z-i}{z+2i}\right|& =& 1\\ & \Rightarrow & \left|z-i\right|=\left|z+2i\right|\end{array}$

This means that $z$ lies on the perpendicular bisector of $(0,1)\text{and}(0,-2)$

$\Rightarrow Img=-\frac{1}{2}$

Now, let

$$\begin{gathered} z=x-\frac{i}{2} \\ \because \left|z\right|=\frac{5}{2} \\ \Rightarrow x^2=6 \end{gathered}$$

$\begin{array}{rcl}\therefore \left|z+3i\right|& =& \left|x+\frac{5i}{2}\right|\\ & =& \sqrt{x^2+\frac{25}{4}}\\ & =& \sqrt{6+\frac{25}{4}}\\ & =& \frac{7}{2}\end{array}$

Hence, option (B) is the correct answer.