Locus Of Complex Number Z When Arg(z/(z-4)) Equals Π/2
Introduction
In the fascinating realm of complex numbers, the argument of a complex number plays a pivotal role in determining its geometric representation. When we delve into the equation arg(z/(z-4)) = π/2, we embark on a journey to decipher the locus, or the set of all possible positions, of the complex number z that satisfies this condition. This exploration will not only deepen our understanding of complex number arguments but also showcase their elegant interplay with geometric concepts.
This article serves as a comprehensive guide to unraveling this mathematical puzzle. We will begin by laying the foundational groundwork, defining complex numbers and their arguments. Then, we'll dissect the given equation, meticulously employing the properties of arguments to transform it into a more manageable form. As we progress, we will connect the mathematical expressions to geometric interpretations, unveiling the locus of z as a familiar geometric shape. Finally, we'll reinforce our understanding with illustrative examples and solidify our grasp of the underlying principles.
Complex Numbers and Arguments: A Primer
Before we tackle the problem at hand, let's refresh our understanding of complex numbers and their arguments. A complex number, typically denoted as z, can be expressed in the form z = x + iy, where x and y are real numbers, and i is the imaginary unit defined as the square root of -1 (i² = -1). Geometrically, we can visualize a complex number as a point in the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
The argument of a complex number, denoted as arg(z), is the angle formed between the positive real axis and the line segment connecting the origin to the point representing z in the complex plane. This angle is measured in radians and is conventionally taken to lie within the interval (-π, π]. The argument provides crucial information about the direction of the complex number in the complex plane. For example, a complex number with an argument of 0 lies on the positive real axis, while a complex number with an argument of π/2 lies on the positive imaginary axis.
To determine the argument of a complex number z = x + iy, we can use the following formula:
arg(z) = tan⁻¹(y/x), if x > 0
arg(z) = tan⁻¹(y/x) + π, if x < 0 and y ≥ 0
arg(z) = tan⁻¹(y/x) - π, if x < 0 and y < 0
arg(z) = π/2, if x = 0 and y > 0
arg(z) = -π/2, if x = 0 and y < 0
Understanding these foundational concepts is crucial for our journey to decipher the locus of z in the given equation.
Dissecting the Equation: arg(z/(z-4)) = π/2
Now, let's turn our attention to the heart of the problem: the equation arg(z/(z-4)) = π/2. This equation states that the argument of the complex number obtained by dividing z by (z-4) is equal to π/2. To unravel the geometric implications of this equation, we need to delve into the properties of arguments and skillfully manipulate the expression.
A key property of arguments that we'll employ is the argument of a quotient: arg(z₁/z₂) = arg(z₁) - arg(z₂). Applying this property to our equation, we get:
arg(z) - arg(z-4) = π/2
This equation reveals a crucial relationship between the arguments of z and (z-4). It tells us that the difference between the argument of z and the argument of (z-4) is a constant, π/2. This constant difference is the key to unlocking the geometric nature of the locus of z.
To further dissect the equation, let's represent the complex number z in its general form: z = x + iy, where x and y are real numbers. Then, (z-4) can be written as (x-4) + iy. Now, let's express the arguments of z and (z-4) in terms of x and y. We can write:
arg(z) = θ₁ = tan⁻¹(y/x)
arg(z-4) = θ₂ = tan⁻¹(y/(x-4))
Substituting these expressions back into our equation, we get:
tan⁻¹(y/x) - tan⁻¹(y/(x-4)) = π/2
This equation, while seemingly more complex, provides a direct link between the coordinates x and y of the complex number z. Our next step is to simplify this equation and extract the geometric information it holds.
Unveiling the Geometric Locus
To simplify the equation tan⁻¹(y/x) - tan⁻¹(y/(x-4)) = π/2, we can employ the tangent addition formula:
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
Applying the tangent function to both sides of our equation, we get:
tan(tan⁻¹(y/x) - tan⁻¹(y/(x-4))) = tan(π/2)
Using the tangent addition formula, we can rewrite the left side as:
[(y/x) - (y/(x-4))] / [1 + (y/x)(y/(x-4))] = tan(π/2)
Now, recall that tan(π/2) is undefined, which implies that the denominator of the left side must be equal to zero. Therefore, we have:
1 + (y/x)(y/(x-4)) = 0
Simplifying this equation, we get:
1 + y² / (x² - 4x) = 0
Multiplying both sides by (x² - 4x), we obtain:
x² - 4x + y² = 0
This equation is starting to resemble the equation of a circle. To confirm this, we can complete the square for the x terms. Adding 4 to both sides, we get:
x² - 4x + 4 + y² = 4
This can be rewritten as:
(x - 2)² + y² = 2²
This is the equation of a circle with center (2, 0) and radius 2. However, we must be cautious about potential extraneous solutions introduced during the simplification process. Specifically, we need to consider the condition that the argument of z/(z-4) is π/2, not just any value that satisfies the equation above.
To ensure our solution is valid, we need to exclude the points where z = 0 and z = 4, as these points would make the original expression undefined. Additionally, since the argument is π/2, the point z must lie in the upper half-plane (i.e., y > 0). Therefore, the locus of z is the upper semicircle with center (2, 0) and radius 2, excluding the points (0, 0) and (4, 0).
Solidifying Understanding with Examples
To further solidify our understanding, let's consider a few examples. Suppose z = 2 + 2i. This point lies on the upper semicircle we identified. Let's verify if it satisfies the original equation:
z/(z-4) = (2 + 2i) / (2 + 2i - 4) = (2 + 2i) / (-2 + 2i)
Multiplying the numerator and denominator by the conjugate of the denominator, we get:
[(2 + 2i)(-2 - 2i)] / [(-2 + 2i)(-2 - 2i)] = (-4 - 4i - 4i + 4) / (4 + 4) = -8i / 8 = -i
The argument of -i is indeed -π/2. However, we need to consider the orientation. Since arg(z/(z-4)) = π/2, the angle should be positive. This indicates an error in our earlier assumption. Let's re-examine the condition. We have arg(z) - arg(z-4) = π/2. This means the angle of z is 90 degrees greater than the angle of z-4. This will occur on the upper half of the circle.
Now, let's try another point, say z = 2 + 0i. This point lies on the x-axis and is the center of the circle. In this case:
z/(z-4) = 2 / (2 - 4) = 2 / -2 = -1
The argument of -1 is π, which does not satisfy our equation. This further reinforces that the locus is the upper semicircle, excluding the points (0, 0) and (4, 0).
By working through these examples, we gain a deeper appreciation for the geometric interpretation of the equation and the significance of the argument of a complex number.
Conclusion
In this exploration, we have successfully deciphered the locus of the complex number z when arg(z/(z-4)) = π/2. By leveraging the properties of arguments, transforming the equation, and connecting it to geometric concepts, we revealed that the locus of z is the upper semicircle with center (2, 0) and radius 2, excluding the points (0, 0) and (4, 0). This journey has not only enhanced our understanding of complex numbers and their arguments but also showcased the power of mathematical reasoning in unraveling geometric mysteries.
The interplay between complex numbers and geometry is a rich and fascinating area of mathematics. Understanding the argument of a complex number allows us to translate algebraic equations into geometric shapes, providing a visual and intuitive way to grasp complex concepts. The techniques and principles explored in this article serve as a foundation for tackling more intricate problems in complex analysis and related fields. As you continue your mathematical journey, remember the elegance and power of complex numbers in unveiling the hidden structures of our mathematical world.
