Animated Solution for Mathematics - Complex Numbers: If $|z-3+2i|\le 4$ then the difference between the greatest value and the least value of $|z|$ is :-

A Journey into the Disk

Welcome, fellow traveler of the complex plane! Today, we are going to unravel a problem that, at first glance, might seem like a dry algebraic exercise, but is actually a beautiful exploration of geometry.

We are looking at the inequality $|z-3+2i|\le 4$. To solve this, we must first stop thinking of $z$ as just a variable and start seeing it as a point in the Argand plane.

Decoding the Region

First, let us rewrite our inequality into the standard form: $|z-(3-2i)|\le 4$. This is the equation of a solid disk.

Think of it as a circular region on a map. The center of this disk, $z_0$, is at $3-2i$, which corresponds to the coordinate point $C(3,-2)$. The radius, $r$, is $4$.

So, imagine a circle with a radius of $4$ units, anchored firmly at $(3,-2)$. Every point $z$ that satisfies this inequality is either on the boundary of this circle or somewhere inside it.

The Origin's Perspective

Our goal is to find the greatest and least values of $|z|$. Geometrically, $|z|$ is simply the distance from the origin $O(0,0)$ to any point $z$ within our disk.

To understand the range of these distances, we must first find how far the center of our disk is from the origin. Using the distance formula, we calculate:

Now, here is the crucial moment of the problem. We have $OC=\sqrt{13}$, which is approximately $3.6$.

Our radius $r$ is $4$. Since $3.6<4$, the distance to the center is less than the radius. This means the origin $O(0,0)$ is not outside the circle; it is tucked safely inside the disk!

Finding the Extremes

With the origin inside the disk, the logic for the maximum and minimum distances becomes intuitive. For the greatest value of $|z|$, we want to be as far from the origin as possible.

Imagine standing at the origin and walking in a straight line through the center $C$ until you hit the far edge of the disk. The distance you travel is the distance to the center plus the radius:

For the least value of $|z|$, we want to be as close to the origin as possible. Since the origin is inside the disk, we can simply stand right on top of it!

If $z$ is at the origin, the distance is $0$. Thus, $|z{|}_{min}=0$.

The Final Step

The problem asks for the difference between the greatest and least values. We have our maximum, $\sqrt{13}+4$, and our minimum, $0$.

Subtracting them gives us $4+\sqrt{13}$.

Isn't that elegant? By visualizing the disk and the origin's position, we bypassed complex algebraic manipulations and arrived at the truth through pure geometry. Keep this visualization in your toolkit—it is the key to mastering complex numbers in the JEE Advanced!