Animated Solution for Mathematics - Complex Numbers: Let $z$ be a complex number such that $|\frac{z-i}{z+2i}|=1$ and $|z|=\frac{5}{2}$. Then the value of $|z+3i|$ is:

Imagine you are standing on the Argand plane, a vast, two-dimensional landscape where every point is a complex number $z=x+iy$. We are embarking on a journey to find the hidden location of a point $z$ that obeys two strict, beautiful laws.

Our first law is given by the equation:

By cross-multiplying, we transform this into $|z-i|=|z-(-2i)|$. This is a profound geometric statement; it tells us that the distance from $z$ to the point $i$ (which is $(0,1)$) is exactly equal to the distance from $z$ to the point $-2i$ (which is $(0,-2)$).

The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them. The midpoint between $(0,1)$ and $(0,-2)$ is $(0,-0.5)$, and since the segment is vertical, the perpendicular bisector must be a horizontal line.

Algebraically, if we set $z=x+iy$, the equation becomes:

Watch as the $x^2$ terms vanish, leaving us with $y^2-2y+1=y^2+4y+4$. Simplifying this, we find $-6y=3$, or:

This horizontal line is the home of our complex number $z$.

Now, we introduce our second law: $|z|=\frac{5}{2}$. This is the equation of a circle centered at the origin with a radius of $\frac{5}{2}$.

In Cartesian terms, this is:

We now have two constraints: our point $z$ must lie on the line $y=-\frac{1}{2}$ and on the circle $x^2+y^2=\frac{25}{4}$. By substituting $y=-\frac{1}{2}$ into the circle equation, we get:

This simplifies to $x^2+\frac{1}{4}=\frac{25}{4}$, which leads us to $x^2=6$.

The question asks for the value of $|z+3i|$. Geometrically, this is the distance between our point $z(x,y)$ and the point $(0,-3)$.

Using the distance formula, we have:

We already know $x^2=6$ and $y=-\frac{1}{2}$. Substituting these values, we get:

Simplifying the term inside the parenthesis, we have $-\frac{1}{2}+3=\frac{5}{2}$. Squaring this gives $\frac{25}{4}$.

Finally, we calculate:

The final result is $\frac{7}{2}$.