Problem 41 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}, z_{2}, z_{3}, z_{4}\) are the four complex numbers represented by the vertices of a quadrilateral taken in order such that \(z_{1}-z_{4}=z_{2}-z_{3}\) and amp \(\frac{z_{4}-z_{1}}{z_{2}-z_{1}}=\frac{\pi}{2}\) then the quadrilateral is a (A) square (B) rhombus (C) rectangle (D) a cyclic quadrilateral

Step-by-Step Solution

Answer

The quadrilateral is a rectangle (Option C).

Step 1: Understand the condition z1-z4=z2-z3

The condition \( z_1 - z_4 = z_2 - z_3 \) implies that the vector \( z_1 \) to \( z_4 \) is equal in magnitude and direction to the vector \( z_2 \) to \( z_3 \). This suggests that the sides of the quadrilateral are equal in opposite pairs, indicating the quadrilateral could be a parallelogram.

Step 2: Analyze the angle condition

The given condition \( \mathrm{amp} \left( \frac{z_4-z_1}{z_2-z_1} \right) = \frac{\pi}{2} \) implies that \( z_4 - z_1 \) and \( z_2 - z_1 \) are perpendicular because the argument between the two complex numbers is \( \frac{\pi}{2} \). In a geometric sense, this indicates a right angle between these sides.

Step 3: Determine the type of quadrilateral

With the properties derived, the quadrilateral has equal opposite sides due to it being a parallelogram (Step 1) and a right angle due to Step 2. In parallelograms, if there is one right angle, then all angles are right angles, thus making it a rectangle.

Key Concepts

Quadrilateral Properties

Quadrilaterals are fascinating shapes in geometry as they have four sides and vertices. When dealing with quadrilaterals in the context of complex numbers, understanding their properties becomes essential. One crucial property is the arrangement and length of their sides.
Quadrilaterals include various types, such as squares, rectangles, rhombuses, and cyclic quadrilaterals. For instance:

• A square has all sides equal and all angles at 90 degrees.
• A rectangle also has all angles at 90 degrees, but only opposite sides are equal.
• A rhombus has all sides equal with opposite angles equal but not necessarily at 90 degrees.
• Cyclic quadrilaterals have opposite angles that sum up to 180 degrees, and they lie on a single circle's circumference.

The conditions explored in complex number problems often help identify these properties by using the unique characteristics of each type. Understanding these will help you classify quadrilaterals effectively when given specific conditions, like in the provided exercise.

Geometric Representation

Complex numbers can be elegantly represented in the geometric plane. Each complex number corresponds to a point, allowing vectors to illustrate their relationship. In the exercise, complex numbers are thought of as points at the vertices of a quadrilateral. The lines joining these points become vectors representing the sides of the quadrilateral.
By looking at the geometric representation:

• The expression \( z_1 - z_4 = z_2 - z_3 \) suggests that the vector from \( z_1 \) to \( z_4 \) is equal to the vector from \( z_2 \) to \( z_3 \), indicating parallel lines or sides.
• This visual insight aids in understanding that the quadrilateral likely exhibits properties of parallelograms, where opposite sides are parallel and equal.

This geometric view is instrumental in visualizing the problem, helping you to think in terms of shapes and symmetry that are naturally tied to the concept of the quadrilateral as represented in a complex plane.

Angles in Complex Numbers

Angles play a pivotal role when dealing with complex numbers, especially in geometric constructions like quadrilaterals. The argument of a complex number is essentially its angle with the positive real axis in the complex plane. When comparing two complex numbers, the argument can describe the angle between them.
In our context:

• The condition \( \mathrm{amp} \left( \frac{z_4 - z_1}{z_2 - z_1} \right) = \frac{\pi}{2} \) shows these two lines are perpendicular since the argument or angle between them is 90 degrees.
• This perpendicularity is a vital clue to determining the quadrilateral's type; when you know two sides are at a right angle, it suggests that the quadrilateral could be a rectangle or square.

Understanding and applying these angle properties in the realm of complex numbers introduces analytical and visual ways to solve geometrical problems involving quadrilaterals.