Question

\(|z+1|+|z-1|=8\)

Short answer

The equation represents an ellipse with foci at -1 and 1, and a major axis length of 8.

Step-by-step solution

Identify the type of problem

This equation involves the absolute values of a complex number. It is a geometric problem in complex numbers.

Understand the geometric interpretation

In the complex plane, \(|z+1|\) and \(|z-1|\) represent the distances from a point z to the points -1 and 1, respectively.

Formulate the geometric condition

The equation \(|z+1| + |z-1| = 8|\) represents an ellipse with the foci at -1 and 1, and the sum of distances from any point on the ellipse to the foci is constant.

Determine the major axis length

The length of the major axis of this ellipse, given by the equation, is 8.

Identify the foci distance

The distance between the foci \(-1\) and \1\ is 2 units.

Verify condition for an ellipse

In an ellipse, the sum of the distances from any point on the ellipse to the foci is equal to the length of the major axis. Hence, the given condition \(|z+1| + |z-1| = 8|\) satisfies this characteristic.

Key concepts

geometric interpretation in complex plane

Complex numbers can be visualized in the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
For instance, the complex number z = a + bi is represented as a point (a, b) on the complex plane. Distances between points in this plane play a crucial role in understanding complex numbers geometrically.
This geometric interpretation helps to simplify complex equations and visualize their solutions more intuitively.

ellipse properties

An ellipse is a geometric shape where the sum of the distances from any point on the ellipse to two fixed points (called foci) is constant.
This unique property translates directly to problems involving complex numbers, like the one in the exercise.
Ellipses have many fascinating properties, including symmetry and the presence of a major and minor axis.
The major axis is the longest diameter passing through both foci, while the minor axis is the shortest diameter, perpendicular to the major axis.

absolute values of complex numbers

The absolute value (or magnitude) of a complex number z = a + bi is denoted as \(|z|\) and is given by \(\sqrt{a^2 + b^2}\).
This represents the distance from the origin to the point (a, b) on the complex plane.
For example, \(|z+1|\) and \(|z-1|\) in the exercise represent the distances from a point z to the points -1 and 1, respectively.
These distances are crucial in defining the ellipse's properties based on the given equation.

foci distance

The foci are the two fixed points used in the definition of an ellipse.
In the exercise, the foci are at points -1 and 1 on the complex plane.
The distance between these two foci can be computed as the distance between -1 and 1, which is 2 units.
This distance between the foci is an essential parameter in describing the shape and dimensions of an ellipse.

major axis of ellipse

The major axis of an ellipse is the longest line segment passing through the center and both foci.
It determines the ellipse's longest dimension.
In the exercise, the major axis is given as 8 units.
This value matches the sum of the distances from any point on the ellipse to the foci, confirming that the equation describes an ellipse.
Understanding the major axis helps in visualizing and graphing the ellipse accurately on the complex plane.