Problem 102 - Algebra and Trigonometry

Question

Describe the graph of all complex numbers with an absolute value of 6

Step-by-Step Solution

Answer

The graph of all complex numbers with an absolute value of 6 is a circle centered at the origin on the complex plane with a radius of 6.

Step 1: Understand the Concept and Define Parameters

Start by understanding the location of complex numbers in the complex plane. The real part, \(a\), correlates to the x-axis, and the imaginary part, \(b\), correlates to the y-axis. In our case, we know that the modulus or absolute value of our complex number is 6.

Step 2: Setting Up the Equation

The absolute value (modulus) of a complex number \(z = a + bi\) is calculated as \(\sqrt{a^2 + b^2}\). Since we are given that the absolute value is 6, we set up the equation \(\sqrt{a^2 + b^2} = 6\). Squaring both sides, we get the equation \(a^2 + b^2 = 36\).

Step 3: Graphing the Solution

The equation \(a^2 + b^2 = 36\) is the equation of a circle centered on the origin with a radius of 6. Thus, the graph of all complex numbers with an absolute value of 6 is a circle centered at the origin on the complex plane with a radius of 6.