Question

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

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

Answer 1

The correct option is B $\frac{7}{2}$

If $\left|\frac{z-i}{z+2i}\right|=1$ and $|z|=\frac{5}{2}$
$\Rightarrow |z-i|=|z+2i|\Rightarrow x^2+(y-1{)}^2=x^2+(y+2{)}^2\Rightarrow y-1=\pm (y+2)\Rightarrow y-1=-y-2\Rightarrow y=-\frac{1}{2}\Rightarrow |z|=\frac{5}{2}\Rightarrow x^2+y^2=\frac{25}{4}\Rightarrow x^2+\frac{1}{4}=\frac{25}{4}\Rightarrow x=\pm \sqrt{6}|z+3i|=\sqrt{x^2+(y+3{)}^2}\Rightarrow |z+3i|=\frac{7}{2}$