Lesson 5.5: Loci and Regions in the Complex Plane
Introduction
Welcome to Lesson 5.5 of Foundation Further Mathematics! In this lesson, we will explore loci and regions in the complex plane. By the end of this lesson, students, you will be able to determine the geometric loci defined by complex conditions and sketch them accurately. 🌍
Learning Objectives
By the end of this lesson, you should be able to:
- Define loci given by $|z - a| = r$ (circles) and $|z - a| = |z - b|$ (perpendicular bisectors).
- Identify loci defined by $\text{arg}(z - a) = \theta$ (half-lines).
- Sketch regions defined by inequalities in $z$.
- Identify and sketch the locus given by a modulus or argument condition.
- Find the Cartesian equation for a specified complex locus.
Loci Based on Distances
1. Circles
The equation for a circle in the complex plane is given by:
$$|z - a| = r$$
where $a$ is the center of the circle and $r$ is the radius. Let’s consider an example:
Example:
If we have the circle defined by $|z - 2 + 3i| = 5$, this means that the center of the circle is at the point $(2, 3)$, and the radius is 5 units. To sketch this, we draw a circle centered at (2, 3) and with a radius of 5.
2. Perpendicular Bisectors
A second important locus in the complex plane is defined by:
$$|z - a| = |z - b|$$
This represents the set of all points equidistant from points $a$ and $b$. This set of points forms the perpendicular bisector of the line segment joining $a$ and $b$.
Example:
Let’s say we have two points $a = 1 + i$ and $b = 3 + 3i$. The perpendicular bisector of these points will be a straight line in the complex plane. To find this line, you first calculate the midpoint and then use the slope to determine the equation of the line.
Loci Based on Angles
3. Half-Lines
Another important locus is defined by the argument of a complex number:
$$\text{arg}(z - a) = \theta$$
This represents all points $z$ in the complex plane that make an angle of $\theta$ with the positive real axis from the point $a$.
Example:
For the locus defined by $\text{arg}(z - (1 + i)) = \frac{\pi}{4}$, we can identify that this locus captures all points $z$ such that the line from $1 + i$ makes a $45^{\circ}$ angle with the positive real axis. This will create a half-line radiating out from (1,1) at that angle.
Sketching Regions from Inequalities
4. Identifying Regions
Sometimes we want to describe regions defined by inequalities. For example, a region defined by $|z - 1| < 2$ represents all points that lie within a circle of radius 2 centered at (1,0).
To sketch this, we can first draw the circle with radius 2 centered at (1, 0) and shade the interior to show that we are including all points inside the circle.
Example:
If we want to sketch the region defined by $|z| > 1$, that represents all points outside the unit circle centered at the origin.
Sketching Given Loci Conditions
5. Sketching Loci
After understanding the different cases, we can combine our knowledge to solve problems that ask for the locus given certain conditions.
Example:
If we need to find the locus defined by $|z - 2| = |z + 2|$, we recognize this is the perpendicular bisector between the points $(2, 0)$ and $(-2, 0)$. A quick sketch shows this will be the vertical line $x = 0$ (the imaginary axis).
Conclusion
In this lesson, students, we have covered how to define and sketch various loci in the complex plane, including circles, perpendicular bisectors, half-lines, and regions defined by inequalities. Understanding these concepts will allow you to visualize complex numbers better and deepen your knowledge of the complex plane. Remember that practice makes perfect!
Study Notes
- Circles: $|z - a| = r$ represents circles in the complex plane.
- Perpendicular Bisectors: $|z - a| = |z - b|$ gives the perpendicular bisector between points $a$ and $b$.
- Half-Lines: $\text{arg}(z - a) = \theta$ describes half-lines in the complex plane.
- Regions from Inequalities: $|z - a| < r$ represents regions inside circles; $|z| > r$ represents regions outside circles.
- Sketching Loci: Combine knowledge to sketch given loci conditions accurately.