Problem 88 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The locus of the complex number \(z\) in an argand plane satisfying the inequality \(\log _{1 / 2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1\left(\right.\) where \(\left.|z-1| \neq \frac{2}{3}\right)\) is (A) a circle (B) interior of a circle (C) exterior of a circle (D) None of these

Step-by-Step Solution

Answer

(C) exterior of a circle.

Step 1: Understanding the Inequality

We are given the inequality \( \log_{1/2}\left( \frac{|z-1|+4}{3|z-1|-2} \right) > 1 \). Since the logarithm is in base \( \frac{1}{2} \), we need to consider how logarithms with bases less than 1 behave. If the log of a number is greater than 1, that means the argument of the log is less than the base raised to 1, which is \( \frac{1}{2}^1 = \frac{1}{2} \). So, our inequality becomes \( \frac{|z-1|+4}{3|z-1|-2} < \frac{1}{2} \).

Step 2: Simplifying the Inequality

We start simplifying \( \frac{|z-1|+4}{3|z-1|-2} < \frac{1}{2} \) by cross-multiplying: \( 2(|z-1| + 4) < 3|z-1| - 2 \). This leads to \( 2|z-1| + 8 < 3|z-1| - 2 \). Rearranging the terms gives \( 8 + 2 < 3|z-1| - 2|z-1| \), which simplifies to \( 10 < |z-1| \).

Step 3: Interpreting the Result

The inequality \( |z-1| > 10 \) describes the exterior of a circle with center at \((1,0)\) and radius 10. The set of points that satisfy this inequality is outside of this circle.

Step 4: Verifying Option

The statement \( |z-1| > 10 \) matches the description: it represents the exterior of a circle. Therefore, the correct answer is (C) exterior of a circle.

Key Concepts

Inequalities in Complex Plane

Inequalities involving complex numbers are examined within the complex plane, a two-dimensional plane representing complex numbers where the horizontal axis corresponds to real numbers and the vertical axis to imaginary numbers. In the given exercise, the inequality describes a condition for the complex number \( z \) in relation to other values. When assessing inequalities in this context, often the absolute value \( |z - a| \) is involved, where \( a \) is a specific complex number or point in the complex plane. The inequality is simplified by considering the geometric implications: the distance from point \( z \) to point \( a \). This inequality then leads us to understand the locus - the set of all points in the complex plane that satisfy the inequality condition. Here, understanding the behavior of logarithms, especially with bases less than 1, is crucial as it affects the interpretation of the inequality in the context of distances in the plane.To solve the given inequality \( \log _{1 / 2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1 \), we first convert this into an inequality of distances or moduli. The solution ultimately leads to a set described by \( |z-1| > 10 \), indicating how inequalities act as boundaries and constraints dictating valid positions for \( z \) in the complex plane.

Argument and Modulus

A complex number \( z = x + yi \) can be characterized by its modulus and argument. The modulus \(|z|\) is the distance from the origin to the point \( z \) in the complex plane, calculated as \( \sqrt{x^2 + y^2} \). The argument, denoted \( \arg(z) \), is the angle made with the positive real axis, often described in radians. In the context of the exercise, modulus is of particular interest since it relates directly to our inequality: \(|z - 1| > 10\). This signifies that the distance from \( z \) to the point \((1,0)\) exceeds 10 units.Modulus provides a shorthand for "distance," simplifying complex number analysis into familiar geometric and algebraic terms. Arguments, while not directly used in this exercise, complete this description by orienting this distance relative to the real axis. Understanding modulus allows us to map out geometric constraints in complex inequality problems, giving more insight into the spatial behavior of \( z \).

Locus of Complex Number

The locus of a complex number refers to the set or path of all points \( z \) fulfilling a particular condition. When given an inequality such as \(|z-a| > r\), it describes the region of the plane where all points are at a distance greater than \( r \) from the point \( a \). For our exercise, the inequality \(|z - 1| > 10\) defines a locus that is the exterior of a circle centered at \((1,0)\) with a radius of 10.This visualization connects algebraic expressions to concrete geometric forms. By understanding loci, one can map complex solutions spatially, offering clarity in interpreting where \( z \) can exist according to specified criteria. Hence, the locus effectively frames potential solutions, giving a visual representation that complements the algebraic understanding.