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Chapter 2: Problem 54

The greatest positive argument of complex number satisfying \(|z-4|=\operatorname{Re}(z)\) is a. \(\frac{\pi}{3}\) b. \(\frac{2 \pi}{3}\). c. \(\frac{\pi}{2}\) d. \(\frac{\pi}{4}\)

Short Answer

Expert verified
The greatest positive argument is \( \frac{\pi}{3} \), option (a).

Step by step solution

01

Define the Complex Number

Let the complex number be denoted by \( z = x + yi \), where \( x \) and \( y \) are real numbers.
02

Set Up the Equation

Given \( |z - 4| = \operatorname{Re}(z) \), substitute \( z = x + yi \). This becomes \( |(x - 4) + yi| = x \).
03

Calculate the Magnitude

The magnitude \( |(x - 4) + yi| \) is calculated as \( \sqrt{(x-4)^2 + y^2} \). So the equation becomes \( \sqrt{(x-4)^2 + y^2} = x \).
04

Simplify the Equation

Square both sides of the equation to eliminate the square root: \( (x-4)^2 + y^2 = x^2 \).
05

Expand and Rearrange

Expand this to get \( x^2 - 8x + 16 + y^2 = x^2 \). Simplify to \( -8x + 16 + y^2 = 0 \). This implies \( y^2 = 8x - 16 \).
06

Find Condition for Validity

Since \( y^2 \) is non-negative, \( 8x - 16 \geq 0 \) giving \( x \geq 2 \).
07

Consider Range of Polar Angle

Considering the geometric interpretation, the equation \( |z-4| = \operatorname{Re}(z) \) suggests a vertical line. The maximum angle, \( \theta \), in the first quadrant from the positive x-axis is \( \tan^{-1}(\frac{y}{x}) \).
08

Determine Greatest Positive Argument

From the derived equation \( y^2 = 8x - 16 \), use \( x \geq 2 \). For greatest positive argument in the first quadrant (aligning with positive directions), when \( x=2 \), \( y = 0 \). So at \( x = 4 \), \( y = 2\sqrt{3} \) gives greatest angle \( \theta = \frac{\pi}{3} \).
09

Check Possible Angles

Re-evaluate for potential angles; as the circle (centered at \(4,0\)) touches the y-axis at \((-4,0)\), the greatest argument arising is indeed from above, confirming \( \frac{\pi}{3} \) alongside typical unit circle conventions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnitude of Complex Numbers
The magnitude of a complex number is like its "distance" from the origin in the complex plane. To find the magnitude of a complex number such as \( z = a + bi \), we use the formula \( |z| = \sqrt{a^2 + b^2} \). This formula is derived from the Pythagorean theorem, as a complex number can be visualized as a right-angled triangle in the complex plane. Here, \( a \) is the real part and \( b \) is the imaginary part.
  • The magnitude tells us how "large" the number is, regardless of its direction or sign.
  • It's always a non-negative value.
In the exercise, calculating the magnitude of \( (x - 4) + yi \) involves substituting these parts into the magnitude formula. This helps in setting up the equations needed to solve for the complex variable \( z \).
Polar Coordinates
Polar coordinates provide a different way to express complex numbers through \( r \) and \( \theta \) where \( r \) is the magnitude and \( \theta \) represents the angle or direction compared to the positive x-axis. This is useful when dealing with rotations or when calculating angles as in our exercise.
  • Use \( r = \sqrt{x^2 + y^2} \) for magnitude.
  • Find \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \) for the angle.
Polar form is especially handy when you're dealing with trigonometric calculations in complex numbers. In the context of this problem, knowing the polar coordinates helps us find the greatest positive argument, which is simply the largest angle \( \theta \) that our complex number can make with the positive x-axis.
Greatest Positive Argument
The argument of a complex number is the angle that the number forms with the positive direction of the x-axis. It's a key part in the polar form of complex numbers. To find the greatest positive argument of a complex number \( z = a + bi \), you evaluate the angle \( \theta \).
  • This involves the angle from \( 0 \) to \( 2\pi \).
  • For our equation, solving \( |z-4| = \operatorname{Re}(z) \) reveals possible values of \( x \) that define our argument.
In the problem, we determined that the greatest positive argument \( \theta \) occurs when \( x = 4 \) and \( y = 2\sqrt{3} \). Substituting these values, the angle \( \theta = \frac{\pi}{3} \) was found to be the greatest possible argument in the first quadrant.
Real and Imaginary Parts
Understanding the real and imaginary parts of a complex number like \( z = a + bi \) is crucial. The real part, \( a \), and the imaginary part, \( b \), allow us to use algebraic methods to manipulate and understand complex numbers.
  • The real part is \( a = \operatorname{Re}(z) \).
  • The imaginary part is \( b = \operatorname{Im}(z) \).
In solving the given problem, recognizing \( \operatorname{Re}(z) \) helped in setting up conditions like \( x \geq 2 \). Understanding these two components allowed for the application of further mathematical operations. This differentiation simplified the problem, allowing us to accurately mark the intersections and angles that define solutions.

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Most popular questions from this chapter

If \(\cos \alpha+2 \cos \beta+3 \cos \gamma=\sin \alpha+2 \sin \beta+3 \sin \gamma=0\), then the value of \(\sin 3 \alpha+8 \sin 3 \beta+27 \sin 3 \gamma\) is a. \(\sin (a+b+\gamma)\) b. \(3 \sin (\alpha+\beta+\gamma)\) c. \(18 \sin (\alpha+\beta+\gamma)\) d. \(\sin (\alpha+2 \beta+3)\)

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