In this explainer, we will learn how to find the loci of a complex equation in the complex plane defined in terms of the argument.
Recall that the argument of a complex number represents the angle that the half-line from the origin to the complex number makes with the positive real axis (-axis), measured in a counterclockwise direction (often referred to as the positive angle or in the positive direction).
For example, given the point , to calculate the argument, we need to consider which of the quadrants of the complex plane the number lies in. In this case, we have a number in the second quadrant. This means that we need to add to the result we get from the inverse tangent. Hence,
Geometrically, this tells us that lies on a ray (or half line) extending radially outwards from the origin making an angle of with the -axis in the counterclockwise direction.
Rather than asking what the argument of a particular complex number is, we could ask the following question: what is the locus of a point of a fixed argument, say, . This represents all the complex numbers which lie on the ray which makes an angle of with the -axis in the counterclockwise direction. Hence, we can simply state that the locus of such that is this ray. However, it is important to remember that since the argument is not defined when , the locus will not include the origin. Note that a ray which does not include its end point is often referred to as an open ray or as an open half-line.
We can of course transform the equation by subtracting a fixed . This will result in a half-line whose endpoint is at in the plane.
Loci Representing Half-Lines in the Plane
The locus of the point such that is a half-line from, but not including, which makes an angle of to the horizontal half-line extending from in the positive -direction measured in a counterclockwise direction.
Example 1: Identifying Loci
Which of the following figures is the correct representation of the locus of which satisfies ?
Answer
We begin by writing the equation of the locus in the standard form to ensure that we correctly identify the value of . Noting that , we rewrite the equation as
By writing it in this form, we can easily identify that . This is the end point of the ray.
Looking at the options, there are three figures in which the endpoint of the ray is at : (a), (b), and (d). Recall that modulus is measured in the counterclockwise direction. Therefore, given that is positive, the answer is not figure (b). We now have two options to consider: (a) and (d). The difference between these two figures is the whether we have used a solid circle, , or a hollow circle, , to represent the end points. Recall that we use the solid circle to indicate that the line includes the endpoint, whereas the hollow circle represents the fact that the end point is not included. Furthermore, recall that the argument of a complex number is not defined for zero. Therefore, the locus of will not include the end point and hence the correct representation for the locus is figure (d).
It is important to remember to use the correct notation for the end points when sketching loci. This is particularly true for exam questions where you may lose marks for not correctly highlighting whether or not the end points are included in your loci.
Example 2: Finding the Equation of a Locus
The figure shows a locus of a point in the complex plane. Write an equation for the locus in the form , where and are constants to be found.
Answer
Recall that for a locus that satisfies , represents the end point of the ray and the directed angle it makes to the horizontal half-line extending from in the positive direction measured in a counterclockwise direction.
From the figure, we can see that the endpoint of the ray is at the point with coordinates . Therefore, the complex number, , that represents this point will be given by . We now consider the angle that the ray makes to the horizontal. The angle given in the figure is measured from the positive horizontal from as expected. However, since the angle represents a clockwise rotation, we should take its negative since the argument is defined in terms of angles measured in the counterclockwise direction. Hence, . Therefore, the equation of the locus is given by
Example 3: Loci
Consider and in the complex plane.
- Find the Cartesian equation of the locus of such that .
- Find the Cartesian equation of the locus of such that .
- Find the point at which the two loci meet and the angle they meet at.
Answer
We can find the Cartesian equation of the locus algebraically or geometrically. We will use the algebraic method for part 1 and the geometric method for part 2.
Part 1
To find the Cartesian equation, we start by substituting into the equation as follows:
Gathering real and imaginary parts, we have
Using the definition of the argument in terms of the tangent function, we can rewrite this as
Notice that we need not worry about which quadrant the complex number is in, since adding or subtracting multiples of to has no effect on the value of . Evaluating the tan on the right-hand side of the equation, we have
Multiplying both sides of the equation by gives
Adding one to both sides and simplifying give us
However, we are not quite finished; we have given the equation of a line and not of a half-line. Therefore, we need to add a condition to either or to restrict our equation to a half-line. We will add our restriction to . We only want the points to the left of . Hence, our condition on is . Note that we use less than and not less than or equal to since the half-line is open; that is, it does not include the endpoint. Therefore, the Cartesian equation of the locus of is given by
Part 2
Recall that a locus in this form is a half-line or ray. Therefore, we will start by finding the slope of the line. Since we are given the angle the line makes to the horizontal, to find the slope , we can simply take its tangent. Hence,
Now we need a point that the line passes through. Using the complex equation of the locus, we can see that the line will pass through the point . Therefore, we can use the slope–intercept form, to write the equation of the line as
Since we know that the locus only represents the half-line, we need to add a condition on or . Hence, the Cartesian equation of the locus of is given by
Part 3
To find the point where the two loci intersect, we can equate the equations we derived in parts 1 and 2 and solve for as follows:
Notice that this equation is only valid for . Therefore, if we find solutions outside of this range, the two loci do not intersect. Gathering the terms on the left-hand side and the constant terms of the right-hand side yields
Hence,
Notice that this solution is in the range . Therefore, the loci intersect. To find the corresponding value of , we can substitute this value of back into the equation we got in part 2 as follows:
Therefore, the two loci intersect at the point . To find the angle that the two loci cross at, we plot them on an Argand diagram.
Using the property of alternative angles, we can see that angle is equal to the difference of the two angles. Hence,
In the previous example, we considered the point of intersection of two loci of the forms and . We saw that the angle that they intersected at was given by . Rather than fixing the value of and , what would happen if we fixed the value of and considered the locus of the point such that and ? Before we consider what this would look like, note that we can rewrite the equation defining this locus. Given that , we can substitute in the value of and as follows:
Using the properties of the argument, we have
We now consider what shape this locus traces out on the complex plane.
As moves in the plane, the is constant. This property might be familiar from the circle theorems of elementary geometry: recall that the angles subtended from an arc are equal if they are in the same segment. The question is whether the converse of this theorem is also true and, in fact, it is. Therefore, we can conclude that the locus of which satisfies the given condition traces out the arc of a circle from point to point . Note that it traces it out in a counterclockwise direction.
Loci Representing Arcs of a Circle
The locus of the point such that is an arc of a circle which subtends an angle of between the points represented by and .
- If , the locus is a major arc.
- If , the locus is a semicircle.
- If , the locus is a minor arc.
Notice that the end points are not part of the locus.
We will now consider an example to demonstrate how we can identify the locus of a point given the equation that it satisfies.
Example 4: Identifying Loci
Which of following figures is the correct representation of the locus of which satisfies ?
Answer
The locus traces out an arc from the points represented by the complex numbers to such that it subtends an angle of . Since , the locus is a major arc. There are only three choices that represent major arcs: (b), (c), and (d). Secondly, we recall that the locus is swept out in an counterclockwise direction from to . This rules out (d) as a possibility. Finally, since the locus does not include the two endpoints, we are looking for the locus that uses two hollow circles, , to represent the end points rather than two solid circles, . Hence, the correct representation for the locus of is figure (b).
Example 5: Finding the Equation of a Locus
The figure shows a locus of a point in the complex plane. Write an equation for the locus in the form , where and are constants to be found.
Answer
Recall that if a locus of a point satisfies , it is a major arc traced out counterclockwise from the point represented by to , which subtends an angle of . Therefore, using the figure, we can immediately see that . We can also easily identify the two endpoints. We need to be careful to ensure that we do not put them in the wrong order. Since the arc it traced out counterclockwise, we can see that the point represents the starting point. Hence, and . Therefore, the equation of the locus is given by
It is possible to use both geometric and algebraic techniques to find the Cartesian equations of loci of this form. In the next example, we will demonstrate geometric methods for finding the equations of loci; then in the following example we will demonstrate algebraic techniques.
Example 6: Finding the Cartesian Equation of a Locus Geometrically
The point satisfies
By plotting the locus on an Argand diagram, find its Cartesian equation.
Answer
The locus is the arc of a circle traced out counterclockwise from the point to the point which subtends an angle of . Since , the locus is a major arc.
To find a Cartesian equation, we need to find the equation of the circle; then we restrict the values of and to exclude the minor arc which it not part of the locus. Therefore, we need to find the coordinates of the center. Notice that since the angle at the center of the circle is a right angle, we can see that it lies directly above the point and on the horizontal that passes through . Hence, its coordinates are . Additionally, we can immediately see that the radius is equal to 6. Therefore, the Cartesian equation of the circle is given by
We now need to restrict the values of and so that they are on the locus. Notice that all the values of and lie above the line passing through and . It is straightforward to see that the equation of this line is . Recall that the locus does not include its end points; therefore, the equation of the locus is given by
Example 7: Finding the Cartesian Equation of a Locus Algebraically
Find the Cartesian equation of the locus of which satisfies
Answer
To find the Cartesian equation of a locus, we start by substituting into the complex equation as follows:
We will now perform the complex division by multiplying the numerator and denominator by the complex conjugate of the denominator as shown:
Expanding the parentheses in the numerator and denominator gives
Gathering together the real and imaginary parts, we have
Hence,
Using the definition of the argument in terms of the tangent function, we can rewrite this as
Notice that we need not worry about which quadrant the complex number is in, since adding or subtracting multiples of to has no effect on the value of . Simplifying, we have
Hence,
We can now complete the square for both and as follows:
Rearranging gives
Finally we need to state the restriction on and to ensure that the points line on the locus, which is the minor arc. Hence, we should only consider the points where .
Example 8: Finding the Properties of a Locus
Given that satisfies by sketching the locus of , find the range of values of and the range of values of the principle argument of .
Answer
Notice that since the value of the argument is , we have a semicircle. Furthermore, it starts from the point and traces out the arc counterclockwise to the point . Hence, the line segment between these two points is the diameter and the length of the radius is half this distance; hence .
Since the radius is 3 and the semicircle lies below the -axis, the range of values of will be from to 0. In particular,
Notice that the inequality with zero is strict since the locus does not include its end points. To find the range of values of the argument of , we need to use some geometric reasoning. Firstly, we draw a ray on the diagram which starts at the origin and is tangent to the semicircle. Using the properties of circles, we know that is a right angle. Furthermore, we know that and . Hence, using trigonometry, we can find as follows:
Hence,
Therefore, the range of the argument of is given by
Notice that the argument is less than zero since the end points are not included in the locus, whereas it includes the argument of the point that is tangent to the circle, so it is greater than or equal to the argument of this point.
Key Points
- We can use the argument in the same way that we can use the modulus to define loci in the complex plane.
- Using our understanding of geometry and the geometry of the complex plane, we can interpret the loci of points which satisfy certain equations and investigate their properties.
- The locus of a point which satisfies is a half-line from, but not including, which makes an angle of to the horizontal half-line extending from in the positive direction. The angle is measured in a counterclockwise direction.
- The locus of a point which satisfies
is an arc of a circle which subtends an angle of between the points
represented by and .
- If , the locus is a major arc.
- If , the locus is a semicircle.
- If , the locus is a minor arc.
