Question

Describe geometrically the set of points in the complex plane satisfying the following equations. $$|z-1+i|=2$$

Short answer

A circle centered at \(1 - i\) with radius 2.

Step-by-step solution

Recognize the Form of the Equation

The given equation is \(|z - (1 - i)| = 2\). In the complex plane, this represents the distance between the point \(z\) and the complex number \(1 - i\).

Interpret the Equation Geometrically

The equation \(|z - (1 - i)| = 2\) means that the distance between any point \(z\) in the set and the point \(1 - i\) is exactly 2.

Identify the Geometric Shape

Any set of points in the complex plane that are at a constant distance from a fixed point forms a circle. Therefore, this equation describes a circle.

Determine the Center and Radius

The center of the circle is the point \(1 - i\) and the radius is 2.

Final Description

The set of points satisfying the equation \(|z - 1 + i| = 2\) is a circle centered at \(1 - i\) with a radius of 2 in the complex plane.

Key concepts

complex numbers

Complex numbers extend the idea of one-dimensional real numbers to two dimensions. They are expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property that \(i^2 = -1\). In the complex plane, the number is represented as a point \((a, b)\).

The horizontal axis, the real axis, represents the real part \(a\) and the vertical axis, the imaginary axis, represents the imaginary part \(b\). A complex number can thus be visualized as a vector from the origin \((0, 0)\) to the point \((a, b)\).

distance formula

The distance formula in the context of complex numbers helps to determine the distance between two points in the complex plane. The distance between two complex numbers \(z_1 = x_1 + y_1i\) and \(z_2 = x_2 + y_2i\) is given by:

\[|z_1 - z_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

In the given exercise, the equation \(|z - (1 - i)| = 2\) describes all points \(z\) that are exactly 2 units away from the point \(1 - i\) in the complex plane. This equation manifests the distance formula applied to complex numbers.

circle in complex plane

A circle in the complex plane is formed by all points that are at a constant distance from a fixed point, known as the center.

For a complex number \(z_0 = x_0 + y_0i\) as the center and a radius \(r\), the equation of the circle is:

\[ |z - z_0| = r \]

In the exercise example, the given equation \(|z - (1 - i)| = 2\) defines a circle with the center at \(1 - i\) and a radius of 2. Every point satisfying this equation lies exactly 2 units away from \(1 - i\), forming a perfect circle.

geometry of complex numbers

Understanding the geometry of complex numbers is crucial for visualizing and interpreting mathematical relationships. The complex plane provides a useful way to translate algebraic operations into geometric transformations.

By identifying that the given equation \(|z - (1 - i)| = 2\) represents a circle, we can easily describe the positional relationship geometrically. The center at \(1 - i\), located at the coordinates \((1, -1)\) in the plane, is the fixed point, and a radius of 2 means every point on this circle is exactly 2 units away from \(1 - i\).

Utilizing geometrical interpretations of complex numbers can simplify and empower problem-solving approaches, bridging the gap between algebra and geometry seamlessly.