Question

Describe geometrically the set of points in the complex plane satisfying the following equations. $$|z+3 i|=4$$

Short answer

A circle centered at (0, -3) with a radius of 4.

Step-by-step solution

- Interpret the equation

Understand the given equation \( |z + 3i| = 4 \). The modulus of a complex number \( z = x + yi \) represents the distance of the point \( (x, y) \) from the origin in the complex plane.

- Translate to geometric terms

Rewrite the modulus equation \( |z + 3i| = 4 \) in terms of \( z \). Since \( z = x + yi \), the expression becomes \( |x + yi + 3i| \). This simplifies to \( |x + (y + 3)i| \).

- Apply modulus definition

The modulus \( |x + (y + 3)i| \) represents the distance from the point \( (x, y + 3) \) to the origin. According to the given equation, this distance is 4.

- Identify the geometric shape

The equation \( |x + (y + 3)i| = 4 \) describes a circle in the complex plane with a center at \( (0, -3) \) and a radius of 4.

- Conclude the geometric description

Conclude that the set of points \( z \) in the complex plane satisfying \( |z + 3i| = 4 \) forms a circle centered at \( (0, -3) \) with a radius of 4.

Key concepts

complex plane

The complex plane is a two-dimensional plane used to represent complex numbers geometrically. Each complex number corresponds to a unique point on this plane. The horizontal axis, known as the real axis, represents the real part of the complex number, while the vertical axis, called the imaginary axis, represents the imaginary part. For example, the complex number \( z = x + yi \) can be plotted as the point \((x, y)\) on the complex plane. Understanding the complex plane helps us visualize and solve equations involving complex numbers more intuitively.

modulus of complex numbers

The modulus of a complex number is a measure of its distance from the origin in the complex plane. Given a complex number \( z = x + yi \), the modulus is denoted as \( |z| \) and calculated using the formula \( |z| = \sqrt{x^2 + y^2} \). This formula is akin to finding the length of the hypotenuse in a right triangle formed by the real and imaginary components of the complex number. For instance, for the complex number in our example \( z + 3i \), we first shift our focus to the point \( (x, y + 3) \) and then compute the modulus, which equates to the Euclidean distance from this point to the origin. When an equation involves the modulus of a complex number, it usually translates to a geometric shape, like a circle, in the complex plane.

geometric interpretation

Geometric interpretation is a powerful method to understand complex number equations. In our exercise, the equation \( |z + 3i| = 4 \) describes a specific set of points. Let's break it down:

When we rewrite it in terms of \( z \), which is \( x + yi \), we get \( |x + (y + 3)i| \). This tells us to look at the points \( (x, y + 3) \). The modulus equation asserts that the distance from these points to the origin (0,0) is 4.

This is the definition of a circle with radius 4, centered at \( (0, -3) \). Thus, any complex number \( z \) that satisfies \( |z + 3i| = 4 \) will lie on this circle. Geometric interpretations like this offer a visual and intuitive grasp of complex number solutions, enhancing our understanding beyond algebraic manipulation.