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Chapter 5: Problem 143

If \(\left|z-\frac{4}{z}\right|=2\), then the maximum value of \(|\mathrm{z}|\) is equal to: [2009] (a) \(\sqrt{5}+1\) (b) 2 (c) \(2+\sqrt{2}\) (d) \(\sqrt{3}+1\)

Short Answer

Expert verified
The maximum value of \(|z|\) is \(\sqrt{5} + 1\), option (a).

Step by step solution

01

Rewrite the Equation

Start with the given equation \( |z - \frac{4}{z}| = 2 \). Substitute \( z = re^{i\theta} \), where \( r = |z| \) and \( e^{i\theta} \) represents the argument. This helps in dealing with the modulus and simplifying the expression.
02

Simplify |z - \frac{4}{z}|

Replace \( z \) with \( re^{i\theta} \) in the expression. The equation becomes \( |re^{i\theta} - \frac{4}{re^{i\theta}}| = 2 \). Simplifying, it turns into \( |r - \frac{4}{r}e^{-i\theta}| = 2 \), but \( e^{-i\theta} \) only affects the argument, not the magnitude. So it is equivalent to \( |r - \frac{4}{r}| = 2 \).
03

Solve for r

Set \( |r - \frac{4}{r}| = 2 \). This means \( r - \frac{4}{r} = 2 \) or \( r - \frac{4}{r} = -2 \). Solve each case separately.
04

Sub-step: Solve r - \frac{4}{r} = 2

Multiply by \( r \) to get \( r^2 - 4 = 2r \). Rearrange to \( r^2 - 2r - 4 = 0 \). Solve using the quadratic formula, \( r = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -2, c = -4 \). This gives \( r = \frac{2\pm\sqrt{20}}{2} = 1\pm\sqrt{5} \).
05

Sub-step: Solve r - \frac{4}{r} = -2

Multiply by \( r \) to get \( r^2 - 4 = -2r \). Rearrange to \( r^2 + 2r - 4 = 0 \). Using the quadratic formula, find \( r = \frac{-2\pm\sqrt{4 + 16}}{2} = -1\pm\sqrt{5} \).
06

Determine the Maximum r

Combine solutions: \( r = 1 + \sqrt{5}, 1 - \sqrt{5} \) from the first case and \( r = \sqrt{5} - 1, -1 - \sqrt{5} \) from the second case. Discard negatives and non-valid based on modulus being positive. The valid positive maximum is \( r = 1 + \sqrt{5} \).
07

Choose the Correct Option

Compare the maximum value \( 1 + \sqrt{5} \) with the given options. This matches option (a) \( \sqrt{5} + 1 \).

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Key Concepts

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

Modulus of a Complex Number
The modulus of a complex number is a fundamental concept in dealing with complex numbers. A complex number is usually expressed in the form of \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. To find the modulus of \( z \), denoted as \( |z| \), we use the formula:
  • \( |z| = \sqrt{a^2 + b^2} \)
This formula gives the distance of the complex number from the origin on the complex plane.
You can think of the modulus as measuring the length of the vector representation of \( z \).
It is always a non-negative number and represents how far \( z \) is from zero in the complex plane.
Polar Form of Complex Numbers
The polar form of a complex number is another way to express complex numbers, outside their standard \( a + bi \) form. It represents a complex number \( z = re^{i\theta} \), where:
  • \( r \) is the modulus of \( z \), or its absolute value.
  • \( \theta \), known as the argument or angle, is the direction the complex number takes from the positive real axis in the complex plane.
This form is particularly useful for multiplication and division of complex numbers, as it simplifies these operations significantly. For example, multiplying two complex numbers in polar form involves multiplying their moduli and adding their arguments:
  • If \( z_1 = r_1 e^{i\theta_1} \) and \( z_2 = r_2 e^{i\theta_2} \), then their product is \( z_1z_2 = r_1r_2 e^{i(\theta_1 + \theta_2)} \).
The polar form is not only elegant but also provides insights into how complex numbers behave in the plane.
Quadratic Equations
Quadratic equations commonly appear when dealing with equations involving modulus or solving expressions for unknowns such as \( |z| \) in complex numbers. The standard form of a quadratic equation is:
  • \( ax^2 + bx + c = 0 \)
To solve quadratic equations, you can use the quadratic formula:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
This formula helps find the roots of the quadratic, where \( a \), \( b \), and \( c \) are coefficients. Solving these kinds of equations can reveal important characteristics, like the maximum or minimum values for a certain set of conditions, such as when calculating the maximum modulus of \( z \). Quadratic equations thus play a pivotal role in delving into deeper complex number analysis.

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