Animated Solution for Mathematics - Complex Numbers: If $z$ is a complex number such that $|z|\ge 1$, then the minimum value of $|z+\frac{1}{2}(3+4i)|$ is:

Welcome, fellow traveler. Today, we are going to explore the beautiful, often misunderstood world of complex numbers. Imagine you are standing on the complex plane, a vast, two-dimensional landscape where every point is a number.

We are given a condition: $|z|\ge 1$. In this landscape, $|z|$ is simply the distance of a point $z$ from the origin $(0,0)$.

So, $|z|=1$ represents the unit circle. The inequality $|z|\ge 1$ tells us that we are restricted to the boundary of this circle and everything outside it. Think of it as a vast, infinite plane with a circular hole cut out of the center.

Seeing the Distance

Now, look at the expression we need to minimize: $|z+\frac{1}{2}(3+4i)|$. At first glance, it looks like a jumble of numbers and variables.

In the language of complex numbers, the absolute value of a difference, $|z-z_0|$, represents the distance between $z$ and a fixed point $z_0$. Our expression is almost in this form. Let's rewrite it:

Now, it is clear! We are looking for the minimum distance between our roaming point $z$ and a fixed point $P=-\frac{1}{2}(3+4i)$.

Let's simplify our fixed point $P$. Distributing the negative half, we get:

In coordinate geometry, this corresponds to the point $(-\frac{3}{2},-2)$. Now, the question becomes: where is this point $P$ in our landscape?

To find out if it lies in our allowed region, we calculate its distance from the origin, $|P|$. Using the distance formula, we have:

Let's perform the arithmetic carefully. The square of $-\frac{3}{2}$ is $\frac{9}{4}$, and the square of $-2$ is $4$. Adding them together, we get:

Taking the square root, we find:

Our fixed point $P$ is $2.5$ units away from the origin.

Here is the moment of truth. Our allowed region is defined by $|z|\ge 1$. Since $2.5\ge 1$, our point $P$ is sitting comfortably in the allowed region!

If you are standing in a field and you want to get as close as possible to a specific spot in that same field, you simply walk to that spot. Since $P$ is in the region, we can choose $z=P$.

The distance $|z-P|$ then becomes $|P-P|=0$. You cannot get a distance smaller than zero. Therefore, the minimum value of the expression is 0.