Animated Solution for Mathematics - Complex Numbers: If $|z+4|\le 3$, then the maximum value of $|z+1|$ is

Welcome, fellow traveler of the complex plane! Today, we are going to demystify one of the most elegant problems in JEE Advanced.

We are looking at the inequality $||z+4||\le 3$.

Do not just see this as a dry algebraic expression. Imagine the Argand plane as a canvas. The expression $||z+4||\le 3$ represents a solid, circular disk centered at the point $-4$ on the real axis, with a radius of $3$. Every point $z$ inside this disk satisfies our condition.

The question asks us to maximize $||z+1||$. In the language of complex numbers, $||z-z_0||$ represents the distance between $z$ and $z_0$.

Therefore, $||z+1||$ is the distance between our variable point $z$ and the fixed point $P(-1,0)$.

Let us examine the distance from the center of our disk, $C(-4,0)$, to our target point $P(-1,0)$. The distance is exactly $3$. This means $P$ is a sentinel standing right on the boundary of our disk.

The challenge now becomes intuitive. If you are standing at point $P$ on the edge of a circular pond, and you want to walk to the point $z$ inside the pond that is furthest away from you, you must walk straight through the center to the diametrically opposite side.

The maximum distance is simply the diameter of the circle. Since the radius is $3$, the diameter is:

Let us verify this using the algebraic titan: the Triangle Inequality. We want to maximize $||z+1||$ given the constraint $||z+4||\le 3$.

We can rewrite the expression as:

By the Triangle Inequality, $||a-b||\le ||a||+||b||$. Setting $a=z+4$ and $b=3$, we obtain:

Since the maximum value of $||z+4||$ is $3$, we substitute this into the inequality:

The geometry and the algebra sing the same song. The maximum value is 6. You have mastered the complex plane today!