Problem 26 - Advanced Engineering Mathematics

Question

Describe the set of points in the complex plane that satisfies \(|z-i|+|z+i|=1\)

Step-by-Step Solution

Answer

There are no points satisfying the condition; the set is empty.

Step 1: Identify the Components of z

Let the complex number be represented as \( z = x + yi \), where \( x \) and \( y \) are real numbers. The expression \( |z-i| \) can be written as \( |x + (y-1)i| \), and \( |z+i| \) can be written as \( |x + (y+1)i| \).

Step 2: Express Distances

Calculate the distance \( |z - i| \) as \( \sqrt{(x-0)^2 + (y-1)^2} \) and \( |z+i| \) as \( \sqrt{(x-0)^2 + (y+1)^2} \). These are the distances from \( z \) to the points \( 0+1i \) and \( 0-1i \), respectively.

Step 3: Setup the Equation

The given condition is \( |z-i| + |z+i| = 1 \). Substitute the distances calculated into this equation:\[ \sqrt{x^2 + (y-1)^2} + \sqrt{x^2 + (y+1)^2} = 1 \]

Step 4: Interpret the Result

The equation \( |z-i| + |z+i| = 1 \) represents the sum of distances from any point \( z \) to the points \( 0+1i \) and \( 0-1i \). This is the property of an ellipse where the total distance from two fixed points (foci) is constant. However, in this case, the sum is less than the distance between the points \( 0+1i \) and \( 0-1i \), which is 2. Thus, there are no points \( z \) that can satisfy this condition, indicating no set exists.

Key Concepts

Complex Numbers

Complex numbers are a fundamental part of mathematics, especially in fields like engineering and physics. A complex number is expressed in the form \( z = x + yi \) where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \).

• The real part of the complex number is \( x \), and is often represented on the horizontal axis of the complex plane.
• The imaginary part is \( yi \), shown on the vertical axis in the complex plane.

The magnitude of a complex number, or its distance from the origin in the complex plane, is given by \( |z| = \sqrt{x^2 + y^2} \). This magnitude can be used to calculate distances between complex numbers and other points in the complex plane, similar to how you would on a coordinate plane. Understanding complex numbers is important for a deeper insight into other complex shapes and functions like ellipses in the complex plane.

Ellipse Properties

Ellipses are a significant geometric shape with intriguing properties. They are defined as a set of points for which the sum of the distances from two fixed points (foci) is constant. In the context of the complex plane, an ellipse is still defined by these properties but mapped using complex numbers.
For example, given two foci in the complex plane at \( 0 + 1i \) and \( 0 - 1i \), an equation of the form \( |z-i| + |z+i| = c \) describes an ellipse when \( c \) exceeds the distance between the foci, here the distance is \( 2 \).

• If \( c = 2 \), it represents the sum of distances equal to the distance between the points, forming what is known as a degenerate ellipse or a line segment.
• For \( c < 2 \), as in the exercise, no points satisfy the equation, meaning no ellipse is formed.

This characteristic of ellipses can expand your understanding of how constraints on complex equations reflect geometric implications.

Distance Formula

The distance formula is a powerful tool in geometry that helps measure the distance between two points. In the context of the complex plane, it takes a similar role. To compute the distance between a complex number \( z = x + yi \) and another point \( a + bi \), the formula adapts as follows:
\[ d = \sqrt{(x-a)^2 + (y-b)^2} \]
This formula remains a cornerstone, especially in describing figures like ellipses. When referring to the condition \( |z - i| + |z + i| = 1 \) in the original exercise, each distance \( |z-i| \) and \( |z+i| \) was expressed using this formula.

• Here, \( |z-i| \) calculates as \( \sqrt{x^2 + (y-1)^2} \), distance from point \((0,1)\) in the complex plane.
• Similarly, \( |z+i| \) is \( \sqrt{x^2 + (y+1)^2} \), distance from point \((0,-1)\).

Using this essential geometric formula not only helps solve problems involving complex numbers but also offers a deeper understanding of their geometric representation, particularly when dealing with figures like ellipses.