Problem 86 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The locus of the complex number \(z\) in an argand plane satisfying the equation $$ \operatorname{Arg}(z+i)-\operatorname{Arg}(z-i)=\frac{\pi}{2} \text { is } $$ (A) boundary of a circle (B) interior of a circle (C) exterior of a circle (D) None of these

Step-by-Step Solution

Answer

(A) boundary of a circle.

Step 1: Identify the Complex Number

Let the complex number be represented as \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part.

Step 2: Rewrite Arguments in Complex Form

The given equation involves arguments of complex numbers. Rewrite them as \( \operatorname{Arg}(x + (y+1)i) \) and \( \operatorname{Arg}(x + (y-1)i) \).

Step 3: Substitute into the Argument Condition

The given condition is \( \operatorname{Arg}(z+i) - \operatorname{Arg}(z-i) = \frac{\pi}{2} \). Substitute the rewritten expressions, resulting in an equation with complex arguments.

Step 4: Interpret the Argument Condition

The condition implies that the difference in the arguments is \( \frac{\pi}{2} \), which is a known condition for lines through certain geometric points.

Step 5: Identify the Geometric Interpretation

Realize that \( \operatorname{Arg}(z + i) - \operatorname{Arg}(z - i) = \frac{\pi}{2} \) represents a semicircle. This condition represents points where the angle subtended by the line segments at \( z \) is \( 90^\circ \).

Step 6: Determine the Locus

The locus of points \( z \) would lie on the boundary of a circle according to the argument condition, which is semicircular from previous steps.

Key Concepts

argand plane

The Argand plane is a visual representation of complex numbers, analogous to the Cartesian plane used for real numbers.

Each point in the Argand plane is a complex number, represented as \( z = x + yi \), where \( x \) is the real component and \( y \) is the imaginary component. Here, the horizontal axis, similar to the x-axis, represents the real part, and the vertical axis, akin to the y-axis, represents the imaginary part.

This plane provides a simple geometric interpretation of complex numbers and their operations, such as addition, multiplication, and finding magnitudes.

• Geometrically, adding two complex numbers involves adding their components graphically in the plane.
• Multiplying involves both dilation and rotation about the origin, giving a clear visual method to understand modulus and argument.

These insights make the Argand plane an essential tool for understanding complex numbers in a way that extends beyond numerical computation to include geometric intuition.

complex number argument

The argument of a complex number, denoted as \( \operatorname{Arg}(z) \), is a fundamental concept in complex analysis. It refers to the angle \( \theta \) made with the positive real axis by the line connecting the origin to the point in the Argand plane.

Mathematically, for a complex number \( z = x + yi \), the argument \( \theta \) is defined as \( \theta = \tan^{-1}\left( \frac{y}{x} \right) \). The argument provides the direction aspect of the complex number, much like how the radius gives magnitude.

The importance of the argument comes into play when comparing phases in electrical engineering or when solving geometric problems in mathspaces such as which we are discussing, involving semicircles and circles:

• It is used to solve equations involving rotations.
• Helps in determining the locus of complex numbers based on angular constraints, like creating semi-circles or other geometric shapes in the Argand plane.

Understanding how the argument compares between different complex numbers, as in the given problem's condition \( \operatorname{Arg}(z+i) - \operatorname{Arg}(z-i) = \frac{\pi}{2} \), helps in interpreting loci of points with specific angular relationships.

geometry of complex numbers

The geometry of complex numbers bridges the gap between algebraic expressions and geometric visuals, bringing a spatial understanding to mathematical problems.

Complex numbers can be interpreted geometrically using the Argand plane, where operations such as addition and multiplication reflect specific transformations.

• Addition corresponds to translating points in the plane, akin to vector addition.
• Multiplication involves scaling magnitudes and rotating the point about the origin.

In additions, specific configurations of complex arguments can form specific geometric shapes.

For example, when the argument of difference between \( z+i \) and \( z-i \) is \( \frac{\pi}{2} \), this setup signifies a semicircle.

The solution provided in the original exercise uses the geometry concept very well. It interprets the condition \( \operatorname{Arg}(z+i) - \operatorname{Arg}(z-i) = \frac{\pi}{2} \) as describing a semicircle.

This type of analysis not only solves algebraic problems but also emphasizes the geometric properties, where the distance from two focuses (points \( i \) and \(-i \)) creates specific structures like circles in geometric terms. This serves to illustrate that complex numbers, far from being an abstract math construct, have profound implications and applications in fields requiring spatial and geometric reasoning.