Let z be a complex number such that |z-iz+2i|=1 and |z|=52.

Question

Let $z$ be a complex number such that $\left|\dfrac{z-i}{z+2i}\right|=1$ and $|z|=\dfrac52$. Then the value of $|z+3i|$ is:

Accepted Answer

Let $z=x+iy$. $|z-i|=|z+2i| \implies -2y+1=4y+4 \implies y=-\dfrac12$.

$|z|=\dfrac52 \implies x^2=\dfrac{25}4-\dfrac14=6$.

$$|z+3i|^2=x^2+\left(\frac52\right)^2=6+\frac{25}4=\frac{49}4 \implies |z+3i|=\frac72.$$

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