Problem 34 - A Complete Resource Book in Mathematics for JEE Main

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: Rewrite the Logarithm Inequality

First, we'll transform the given logarithmic inequality to a manageable form.The inequality is given by:\[\log_{1/2}\left(\frac{|z-1|+4}{3|z-1|-2}\right) > 1\]We can rewrite this inequality using the property of logarithms:\[\frac{|z-1|+4}{3|z-1|-2} < \left(\frac{1}{2}\right)^1 = \frac{1}{2}\] This means \[2(|z-1|+4) < 3|z-1|-2\]

Step 2: Simplify the Inequality

Simplify the inequality obtained from Step 1:\[2|z-1| + 8 < 3|z-1| - 2\]Rearranging the terms yields:\[8 + 2 < 3|z-1| - 2|z-1|\]Solving this gives:\[10 < |z-1|\]Thus, \[|z-1| > 10\]

Step 3: Interpret the Inequality Geometrically

The inequality \(|z-1| > 10\) represents the set of all points in the Argand plane that are exterior to a circle with center at \(1\) and radius \(10\).This means any complex number \(z\) which is further than 10 units away from the point \(1\) (on the complex plane) is part of the solution.

Key Concepts

Argand plane

The Argand plane is a useful tool in complex number theory. It is a two-dimensional plane where we visually represent complex numbers.
Each complex number can be written in the form \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit. In the Argand plane, the real part \(x\) is plotted along the horizontal axis, called the real axis, and the imaginary part \(y\) is plotted along the vertical axis, called the imaginary axis.
This makes it easy to see and understand the geometric properties of complex numbers. For instance, the modulus (or absolute value) of a complex number is visualized as the distance from the origin \((0,0)\) to the point \((x,y)\) on the plane. This distance can be calculated using the formula \(|z| = \sqrt{x^2 + y^2}\). The Argand plane is also crucial for visualizing the solutions to various equations and inequalities given in the complex plane.

logarithmic inequality

Logarithmic inequalities involve expressions that include a logarithm. Solving these inequalities often involves understanding the properties of logarithms.
One key property is the change of base formula, which allows us to adjust the base of the logarithm to a more convenient number. In the original exercise, the inequality \(\log_{1/2}\left(\frac{|z-1|+4}{3|z-1|-2}\right) > 1\) is transformed using the property that \(a < b\) whenever \(\log_{c}(a) > \log_{c}(b)\) if the base \(c\) is between 0 and 1.
When bases of logarithms are fractions, as in our problem, smaller outputs indicate larger inputs due to the base being less than one. This reversal of inequality is a crucial insight that allows us to turn the inequality into a more typical algebraic form, ultimately leading to understanding the locus as a geometric set.

geometric interpretation of inequalities

Geometric interpretation provides a visual method to understand solutions to complex inequalities in the Argand plane.
The inequality \(|z-1| > 10\) is interpreted geometrically to identify a locus. The expression \(|z-1|\) represents the distance of a point \(z = x + yi\) from the point \(1 + 0i\) on the Argand plane. Hence, \(|z-1| > 10\) describes all points \(z\) that lie outside a circle centered at \(1+0i\) with a radius of 10 units.
Therefore, in the complex plane, the locus of points satisfying this inequality is the region outside this circle. Understanding this geometric interpretation helps grasp how inequalities define boundaries and regions in the Argand plane.