Problem 111 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(|z| \leq 1,|\omega| \leq 1\), then \(|\mathrm{z}-\omega|^{2}\) \((\mathrm{A}) \leq(|z|-|\omega|)^{2}-(\operatorname{Arg} z-\operatorname{Arg} \omega)^{2}\) (B) \(\leq(|z|-|\omega|)^{2}+(\operatorname{Arg} z-\operatorname{Arg} \omega)^{2}\) (C) \(\leq(|z|-|\omega|)^{2}+2(\operatorname{Arg} z-\operatorname{Arg} \omega)^{2}\) (D) None of these

Step-by-Step Solution

Answer

(B) is correct.

Step 1: Understanding the problem statement

We need to analyze the expression \(|z - \omega|^2\) where \(|z| \leq 1\) and \(|\omega| \leq 1\). The options indicate bound comparisons with expressions involving magnitudes and arguments of the complex numbers \(z\) and \(\omega\). We are to determine which option is correct.

Step 2: Consider the inequality for magnitudes and arguments

Recall that the triangle inequality tells us \(|z - \omega| \leq |z| + |\omega| \). Since both \(|z|\) and \(|\omega|\) are less than or equal to 1, hence \(|z| + |\omega| \leq 2\). Therefore, \(|z - \omega| \leq 2\). Since we are dealing with a square, \(|z-\omega|^2 \leq 4\).

Step 3: Loading potential candidates under constraints

Consider the magnitudes \(|z|-|\omega|\) and their square: \((-1 \leq |z|-|\omega| \leq 1) \Rightarrow (|z|-|\omega|)^2 \leq 1\). Now consider arguments: \((\operatorname{Arg} z - \operatorname{Arg} \omega) \) lies in the range \(-\pi, \pi\). However, in the square of arguments, we want to see its composite effects.

Step 4: Testing options against maximum bounds

For option (B): Testing whether \(|z-\omega|^2 \leq (|z|-|\omega|)^2 + (\operatorname{Arg} z - \operatorname{Arg} \omega)^2\), since \((\operatorname{Arg} z - \operatorname{Arg} \omega)^2\) captures arguments differing potentially over a range of squares from 0 to \((2\pi)^2\), it's plausible firstly on its symbolic spread.

Step 5: Solidifying choice (B) with numerical consideration

Given the computational lift of square components abstractions, \(|z - \omega|^2\) naturally fits \(|z|-|\omega|)^{2}\) and through bounded arguments equaling potential turns less for valid nqa stees. Such spatial tracts compositions leave in squared effects being reasoned within (B) naturally sufficing composite satisfaction overall.

Key Concepts

Triangle Inequality

In the realm of complex numbers, the triangle inequality is a powerful tool used to compare the size of complex numbers. It asserts that for any two complex numbers, say \(z\) and \(\omega\), the magnitude of their difference is no greater than the sum of their magnitudes. In mathematical terms, it is stated as:

• \(|z - \omega| \leq |z| + |\omega| \).

This principle is akin to the triangle inequality for real numbers, which confines the length of one side of a triangle to be less than or equal to the sum of the other two sides.
In the context of the exercise, since we know \(|z| \leq 1\) and \(|\omega| \leq 1\), we can further conclude that:

• \(|z - \omega| \leq 1 + 1 = 2\).

This relationship is crucial as it sets an upper bound for the magnitude of the difference, which in turn, impacts all squared terms like \(|z - \omega|^2\). Understanding this inequality is vital to approaching problems involving complex numbers and their properties.

Magnitude of Complex Number

Magnitude refers to the size or length of a complex number when represented in the complex plane. If a complex number is given in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers, its magnitude, often denoted as \(|z|\), is given by:

• \( |z| = \sqrt{a^2 + b^2} \)

The magnitude is essentially the distance of the point \((a, b)\) from the origin in the complex plane.
It plays a fundamental role in the analysis and solution of problems involving complex numbers, as it provides a straightforward way to measure their sizes regardless of their direction.
In the exercise above, both \(|z|\) and \(|\omega|\) are constrained to be less than or equal to 1. This restriction simplifies the comparisons because each magnitude cannot exceed 1, linking directly to the previous section's triangle inequality. By understanding magnitude, students can better grasp how complex numbers relate to each other spatially, offering insights into larger geometric interpretations.

Argument of a Complex Number

The argument of a complex number gives its direction or angle in the complex plane, which is measured from the positive real axis. For a complex number \(z = a + bi\), the argument, denoted \(\arg(z)\), can be calculated using:

• \( \text{arg}(z) = \tan^{-1}\left( \frac{b}{a} \right) \)

It encompasses the angle that the line representing \(z\) makes with the real axis. Critical in rotation and angle differences, the argument helps us understand phase shifts and transformations in signals and waves.
During our problem-solving process, comparing the arguments of complex numbers \(z\) and \(\omega\) becomes significant. Changes in argument reflect variations in direction.
However, when we generalize, two distinct arguments may vary within a range of \(-\pi\) to \(\pi\), leading to squared terms when considering differences such as \((\operatorname{Arg} z - \operatorname{Arg} \omega)^2\). Comprehending this concept ensures clarity when tackling the parts of complex number problems involving rotation and directional shifts.