Solving the Equation arg(z/(z-2)) π/2: A Comprehensive Guide

Understanding how to solve complex number equations, particularly those involving the argument (arg) of a complex number, is a fundamental skill in advanced mathematics and physics. In this article, we will explore the solution for the equation arg(z/(z-2)) π/2. We will break down the solution step-by-step and provide a geometric interpretation to enhance comprehension.

Step-by-Step Solution

Let's begin by understanding the implications of the condition arg(z/(z-2)) π/2. This condition indicates that the complex number z/(z-2) lies on the positive imaginary axis. This occurs when the real part of the fraction is 0 and the imaginary part is positive.

Step 1: Express z in Terms of x and y

Let z x yi, where x and y are real numbers.

Step 2: Write the Equation

Substitute z into the given equation:

$$ arg left frac{z}{z-2} right frac{pi}{2} $$
$$ frac{x yi}{x - 2 yi} a bi $$
$$ a bi frac{(x yi)(x - 2 - yi)}{(x - 2 yi)(x - 2 - yi)} $$
$$ a bi frac{x(x - 2) - y^2 yi(x - 2) xyi^2}{(x - 2)^2 y^2} $$
$$ a bi frac{(x^2 - 2x - y^2) yi(x - 2)}{(x - 2)^2 y^2} $$
$$ a bi frac{x^2 - 2x - y^2}{(x - 2)^2 y^2} i left( frac{x - 2}{(x - 2)^2 y^2} right) $$

Step 3: Set Real Part to Zero

For arg(z/(z-2)) π/2, the real part must be zero:

$$ x^2 - 2x - y^2 0 $$
$$ y^2 x^2 - 2x $$
$$ y^2 2x - x^2 $$

Step 4: Solve the Real Part Equation

This is a quadratic equation in x and can be solved for x:

$$ y^2 2x - x^2 $$
$$ x^2 - 2x y^2 0 $$
$$ x^2 - 2x 1 - 1 y^2 0 $$
$$ (x - 1)^2 - 1 y^2 0 $$
$$ (x - 1)^2 y^2 1 $$

The solutions for x and y lie on a circle centered at (1, 0) with a radius of 1.

Step 5: Find the Range of x

The quadratic y^2 2x - x^2 must be non-negative:

$$ 2x - x^2 ≥ 0 $$
$$ -x(x - 2) ≥ 0 $$
$$ x(x - 2) ≤ 0 $$

This inequality holds for x in [0, 2].

Step 6: Find Corresponding y

For each x in the interval [0, 2]:

$$ y ± sqrt{2x - x^2} $$

Step 7: Combine Solutions

Thus, the solutions for z are:

$$ z x isqrt{2x - x^2} $$
$$ z x - isqrt{2x - x^2} $$
$$ x in [0, 2] $$

Geometric Interpretation

Alternatively, we can provide a geometric interpretation to understand the solution. Let's geometrically interpret the term inside arg. Taking a point z on the complex plane, translate it 2 units to the right, then invert with respect to a circle of radius √2 centered at the origin, reflect that with respect to the x-axis, and finally translate 1 unit to the right. The locus of points that satisfy the given argument condition will be a semicircle on the positive imaginary axis.

Geometric Steps

Translate the point z 2 units to the the translated point with respect to a circle of radius √2 centered at the the inverted point with respect to the the reflected point 1 unit to the right.Ensure the point lies on the positive imaginary axis to satisfy the argument condition.

The locus of points that satisfy the given condition is a semicircle on the positive imaginary axis with a radius of √2 and centered at (1, 0).

Conclusion

The complete solutions to the equation arg(z/(z-2)) π/2 are:

$$ z x isqrt{2x - x^2} $$
$$ z x - isqrt{2x - x^2} $$
$$ x in [0, 2] $$