If |z + 4| ≤ 3, then the maximum value of |z + 1| is
6
Understanding the Complex Number Problem
The question asks for the maximum value of \(|z + 1|\) given the condition \(|z + 4| \le 3\). Here, \(z\) is a complex number.
Let's break down the terms:
- \(|z + 4|\) represents the distance between the complex number \(z\) and the point \(-4\) in the complex plane.
- The inequality \(|z + 4| \le 3\) means that the distance between \(z\) and \(-4\) is less than or equal to 3. Geometrically, this describes a closed disk (a circle including its interior) centered at \(-4\) with a radius of 3. Any complex number \(z\) satisfying the condition lies within or on this circle.
- \(|z + 1|\) represents the distance between the complex number \(z\) and the point \(-1\) in the complex plane. We want to find the largest possible value of this distance for any \(z\) that lies within the disk \(|z + 4| \le 3\).
Finding the Maximum Distance Geometrically
We need to find the point \(z\) within or on the disk centered at \(-4\) with radius 3 that is farthest away from the point \(-1\).
Consider the points on the real axis in the complex plane:
- Center of the disk: \(-4\)
- Radius of the disk: \(3\)
- Point from which we measure distance: \(-1\)
The points inside or on the disk are those \(z\) such that their distance from \(-4\) is at most 3. The points on the circumference are at a distance exactly 3 from \(-4\). These points range from \(-4 - 3 = -7\) to \(-4 + 3 = -1\) on the real axis.
The point \(-1\) is located on the boundary of the disk, as the distance between the center \(-4\) and \(-1\) is \(|-4 - (-1)| = |-3| = 3\).
To maximize the distance \(|z + 1|\) (distance from \(z\) to \(-1\)), we should pick a point \(z\) within the disk that is as far as possible from \(-1\). This point will lie on the line passing through the center \(-4\) and the point \(-1\). Since \(-1\) is on the boundary, the point farthest from \(-1\) within the disk will be the point on the circumference on the opposite side of the center \(-4\) relative to \(-1\).
The point on the circumference farthest from \(-1\) is the point on the real axis to the left of \(-4\) at a distance of 3 from \(-4\). This point is \(-4 - 3 = -7\).
Let's verify if \(z = -7\) is within the allowed region:
\(|z + 4| = |-7 + 4| = |-3| = 3\). Since \(3 \le 3\), \(z = -7\) is on the boundary of the disk.
Now, let's calculate \(|z + 1|\) for \(z = -7\):
\(|z + 1| = |-7 + 1| = |-6| = 6\).
This suggests that the maximum value is 6.
Finding the Maximum Value Using Triangle Inequality
We can use the triangle inequality, which states that for any complex numbers \(a\) and \(b\), \(|a + b| \le |a| + |b|\).
We want to maximize \(|z + 1|\) subject to \(|z + 4| \le 3\).
We can rewrite \(z + 1\) in terms of \(z + 4\):
\(z + 1 = (z + 4) - 3\)
Now, apply the triangle inequality to \(|(z + 4) - 3|\):
\(|(z + 4) - 3| \le |z + 4| + |-3|\)
We are given \(|z + 4| \le 3\), and we know \(|-3| = 3\).
So, \(|z + 1| \le 3 + 3\)
\(|z + 1| \le 6\)
This inequality shows that the maximum possible value for \(|z + 1|\) is 6. This maximum is achieved when the complex numbers \(z + 4\) and \(-3\) point in the same direction. If \(z + 4 = k(-3)\) for some positive real number \(k\), then \(|z + 4| = |k(-3)| = k|-3| = 3k\). Given \(|z + 4| = 3\) for the maximum case, we get \(3 = 3k\), so \(k = 1\). This implies \(z + 4 = -3\), or \(z = -7\), which is the same point we found using geometrical reasoning.
Conclusion
Both the geometrical approach and the triangle inequality method show that the maximum value of \(|z + 1|\) when \(|z + 4| \le 3\) is 6.
| Concept | Description |
|---|---|
| \(|z - w|\) | Distance between complex numbers \(z\) and \(w\) in the complex plane. |
| \(|z + 4| \le 3\) | Region of points \(z\) whose distance from \(-4\) is less than or equal to 3 (closed disk centered at \(-4\) with radius 3). |
| \(|z + 1|\) | Distance between point \(z\) and point \(-1\). |
| Triangle Inequality | \(|a + b| \le |a| + |b|\) for complex numbers \(a, b\). Used to find bounds. |
Revision Table: Maximum Value of Modulus
| Given | To Find | Method | Maximum Value | Point \(z\) for Max |
|---|---|---|---|---|
| \(|z + 4| \le 3\) | Maximum of \(|z + 1|\) | Geometry & Triangle Inequality | 6 | \(z = -7\) |
Additional Information: Complex Number Geometry
The modulus of a complex number, \(|z|\), represents its distance from the origin (0, 0) in the complex plane. The expression \(|z - a|\) represents the distance between the complex number \(z\) and the complex number \(a\).
Inequalities involving modulus often describe geometric shapes:
- \(|z - a| = r\): Represents a circle centered at \(a\) with radius \(r\).
- \(|z - a| \le r\): Represents a closed disk (circle and its interior) centered at \(a\) with radius \(r\).
- \(|z - a| < r\): Represents an open disk (interior of a circle) centered at \(a\) with radius \(r\).
- \(|z - a| > r\): Represents the region outside the circle centered at \(a\) with radius \(r\).
- \(|z - a| \ge r\): Represents the region outside or on the circle centered at \(a\) with radius \(r\).
- \(|z - a| = |z - b|\): Represents the perpendicular bisector of the line segment joining \(a\) and \(b\).
These geometric interpretations are very useful for solving problems involving inequalities and maximum/minimum values of complex number expressions.
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- What is the real part of the number $5i$?
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