Question

Describe geometrically the set of points in the complex plane satisfying the following equations. $$|z+1|+|z-1|=8$$

Short answer

The set of points is an ellipse with foci at \(-1\) and \(1\), and the major axis length is 8.

Step-by-step solution

Identify the Problem

The equation to describe geometrically is \(|z + 1| + |z - 1| = 8\). We need to interpret this in the complex plane.

Recall the Definition of Absolute Value

In the complex plane, \(|z - a|\) represents the distance between the point \(z\) and the point \(a\). Similarly, \(|z + 1|\) is the distance between \(z\) and \(-1\), and \(|z - 1|\) is the distance between \(z\) and \(+1\).

Interpret the Given Equation

The equation \(|z + 1| + |z - 1| = 8\) can be interpreted as the sum of distances from any point \(z\) on the complex plane to the points \(-1\) and \(1\) being 8.

Understand Geometry of Equation

In geometry, the sum of distances from any point on an ellipse to its two foci is constant. For this problem, the points \(-1\) and \(1\) are the foci of an ellipse, and the constant sum is 8.

Match Given Equation to Ellipse Form

Therefore, \(|z + 1| + |z - 1| = 8\) describes an ellipse with foci at \(-1\) and \(+1\) on the complex plane and a total distance of 8.

Geometric Description

The set of points in the complex plane satisfying the given equation is an ellipse with foci at \(-1\) and \(1\), with the major axis length being 8.

Key concepts

Absolute Value in Complex Plane

The concept of absolute value in the context of the complex plane is a fundamental topic in complex analysis. In simple terms, the absolute value (or modulus) of a complex number represents its distance from the origin on the complex plane.

Consider a complex number denoted as \(z\) which can be expressed as \(z = x + iy\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit. The absolute value of \(z\) is given by:
\[|z| = \sqrt{x^2 + y^2}\]
In a geometric sense, this formula calculates the Euclidean distance between the point \((x, y)\) and the origin \((0, 0)\).

For a more general form, let's consider the distance from any point \(z\) to another point \(a\) in the complex plane. This distance is represented as \(|z - a|\).

For example:

• \(|z + 1|\) is the distance between the point \(z\) and \(-1\).
• \(|z - 1|\) is the distance between the point \(z\) and \(+1\).

Understanding and visualizing absolute value in the complex plane is key to interpreting many geometrical problems.

Ellipse with Foci

An ellipse is a critical concept in geometry and is defined by its unique property of foci. By definition, an ellipse is the set of all points for which the sum of the distances to two fixed points (the foci) is constant.

In the given problem, we need to interpret the equation:\[ |z + 1| + |z - 1| = 8 \] This equation tells us that for any point \(z\) in the complex plane, the sum of its distances to the points \(-1\) and \(1\) is always equal to \(8\).

• Here, \(-1\) and \(1\) act as the foci of the ellipse.
• The constant sum of distances, which is \(8\), is larger than the distance between the two foci.

For an ellipse centered at the origin with foci at \(-a\) and \(a\) on the real axis, the total sum of the distances from any point on the ellipse to the two foci is always twice the length of the major semi-axis. In other words, if the distance between the two foci is \(2c\), the constant sum is \(2a\) where \(a > c\).

This property is central to solving and understanding complex plane geometry problems involving ellipses.

Geometric Interpretation of Equations

Interpreting equations geometrically in the complex plane can turn abstract algebraic concepts into visual and intuitive understandings. Consider the primary equation:
\[ |z + 1| + |z - 1| = 8 \] We need to convert this algebraic statement into a geometric understanding. Here:

• \(|z + 1|\) is the distance from any point \(z\) to \(-1\).
• \(|z - 1|\) is the distance from any point \(z\) to \(1\).
• The sum of these distances is constant and equal to \(8\).

This equation represents an ellipse with foci at \(-1\) and \(1\) and a total distance across these points of \(8\). The principle behind this is rooted in the definition of an ellipse: a shape where the sum of the distances to the foci is constant.

Understanding these geometric interpretations in equations helps solve and visualize complex problems, turn equations into more tangible concepts, and deeply grasp different geometrical structures like ellipses, hyperbolas, and parabolas.