# Complex Numbers Locus Question

• Amaru58
In summary, to find the cartesian equation of the locus |z+3+2i|=Re(z), we let z = x + iy and substitute into Re(z) = x, giving us x = \left| x + 3 + iy + 2i\right|. Simplifying, we get x^2 = x^2 + 6x + 9 + y^2 + 4y + 4, which can be written in the canonical form f(x,y) = 0 as 6x + 4y + y^2 = -13.
Amaru58

## Homework Statement

Find the cartesian equation of the locus |z+3+2i|=Re(z)

## The Attempt at a Solution

You let z= x +iy
therefore Re(z)=x?

Amaru58 said:

## Homework Statement

Find the cartesian equation of the locus |z+3+2i|=Re(z)

## The Attempt at a Solution

You let z= x +iy
therefore Re(z)=x?
Indeed. So you now have

$$x = \left| x + 3 +iy + 2i\right|$$

Can you write that in canonical form?

OK. I'm not too sure what you mean by canonical form?
Would x = {(x+3)^2 + (y+2)^2}^1/2
Therefore: x^2 = x^2 +6x + 9 +y^2 +4y+4
6x+4y+y^2=-13
I'm not quite sure what to do now
Thanks for the help

Amaru58 said:
OK. I'm not too sure what you mean by canonical form?
Would x = {(x+3)^2 + (y+2)^2}^1/2
Therefore: x^2 = x^2 +6x + 9 +y^2 +4y+4
6x+4y+y^2=-13
I'm not quite sure what to do now
Thanks for the help
You're there. The canonical form in Cartesian coordinates is usually f(x,y) = 0 or f(x,y) = const., which is what you have there.

## 1. What is a complex number locus?

A complex number locus is a set of points on a complex plane that satisfy a specific condition or equation. It is similar to a graph in the real number system, but instead of plotting points on a horizontal and vertical axis, complex numbers are plotted on a horizontal and vertical axis representing the real and imaginary parts respectively.

## 2. How do you graph a complex number locus?

To graph a complex number locus, you first need to determine the equation or condition that the points must satisfy. Then, plot the points on a complex plane and connect them to form a shape. You can also use technology, such as a graphing calculator, to help you graph the locus.

## 3. What are some common examples of complex number loci?

Some common examples of complex number loci include circles, ellipses, and hyperbolas. These shapes can be described by equations involving complex numbers, such as z = a + bi, where a and b are real numbers and i is the imaginary unit.

## 4. How are complex number loci used in real life?

Complex number loci have many real-life applications, particularly in science and engineering. They are used to model and analyze electrical circuits, fluid flow, and other physical systems. They also have applications in computer graphics, cryptography, and signal processing.

## 5. What is the significance of the complex number locus question?

The complex number locus question is significant because it helps us understand the behavior and properties of complex numbers. By studying and graphing complex number loci, we can gain insights into the relationships between real and imaginary numbers and how they interact with each other. This knowledge is essential in many fields of mathematics and science.

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