Animated Solution for Mathematics - Complex Numbers: If $\left|\frac{z-i}{z+2i}\right|=1$, $|z|=\frac{5}{2}$ then value of $|z+3i|$ is

A Journey Beyond Algebra

Welcome, future engineers! Today, we are not just solving a problem; we are exploring the landscape of the Argand plane. Many students approach complex numbers as a dry exercise in algebra—a tedious dance of $x$ and $y$.

But I want you to see the beauty hidden beneath the surface. When you see

do not immediately reach for the pen to square everything. Pause. Breathe. Visualize.

This equation is a statement of distance. It tells us that the distance from $z$ to $i$ is exactly equal to the distance from $z$ to $-2i$. In the language of geometry, the locus of points equidistant from two fixed points is the perpendicular bisector of the segment connecting them.

The Locus of Equilibrium

Let us find this line. Our two points are $z_1=i$ and $z_2=-2i$. These points lie on the imaginary axis.

The midpoint between $i$ and $-2i$ is

Since the points are on the vertical axis, the perpendicular bisector must be a horizontal line passing through this midpoint. Thus, our locus is the line $y=-\frac{1}{2}$.

Algebraically, if we set $z=x+iy$, we are essentially saying that for any $z$ satisfying our condition, the imaginary part must be $-\frac{1}{2}$. We have just constrained our complex number to a single, infinite horizontal line.

The Circle of Constraint

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

We are now looking for the intersection of the line $y=-\frac{1}{2}$ and the circle

This is where the magic happens. We don't need to find the exact coordinates of $z$. We only need the intersection points.

By substituting $y=-\frac{1}{2}$ into the circle equation, we get

This simplifies to

Subtracting $\frac{1}{4}$ from both sides, we find $x^2=\frac{24}{4}=6$. Notice how we stopped at $x^2$. We did not waste time calculating $x=\pm \sqrt{6}$. In competitive exams, the most dangerous trap is doing unnecessary work.

The Final Destination

We are asked to find $|z+3i|$. Geometrically, this is the distance between our point $z$ and the point $-3i$.

Using our algebraic form $z=x+iy$, this distance is

We already know $x^2=6$ and $y=-\frac{1}{2}$. Let us substitute these values into our target expression:

The term inside the parenthesis is $\frac{5}{2}$. Squaring it gives $\frac{25}{4}$. So, we are left with

Combining these, we get

The final result is $\frac{7}{2}$. Look at that elegance! The numbers collapse perfectly. This is the reward for understanding the geometry before diving into the algebra. You didn't just solve a problem; you navigated a coordinate system. Keep this mindset, and no complex number problem will ever intimidate you again.