Problem 155 - 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.

Step 1: Identify Geometric Property

Since \(z_1 - z_4 = z_2 - z_3\), the opposite sides of the quadrilateral are equal. This is a property of a parallelogram.

Step 2: Analyze Argument Condition

The given condition \(\text{amp}\left(\frac{z_4-z_1}{z_2-z_1}\right)=\frac{\pi}{2}\) implies that the vector \(z_4-z_1\) is perpendicular to the vector \(z_2-z_1\). When two consecutive vectors are perpendicular, it suggests that the angle between these vectors is 90 degrees.

Step 3: Infer Quadrilateral Type

A quadrilateral that is both a parallelogram and has a right angle is a rectangle. Since both \(z_1 - z_4\) and \(z_2 - z_1\) are perpendicular, indicating right angles between consecutive sides, the quadrilateral must be a rectangle, not a square or just any parallelogram.

Key Concepts

Quadrilateral

A quadrilateral is a four-sided polygon. It has four vertices and four edges. Types of quadrilaterals include squares, rectangles, rhombuses, and more. Each has its unique properties, but they all share the common feature of having four sides. When dealing with complex numbers as points in the plane, each vertex of the quadrilateral can represent a complex number.
For a general quadrilateral formed by complex numbers:

• If complex numbers represent the vertices, the sides of the quadrilateral are the differences of these numbers.
• The shape of the quadrilateral depends on these differences and their relationships.

Thus, the properties of a quadrilateral relate directly to the geometric properties of complex numbers representing its sides.

Parallelogram

A parallelogram is a special type of quadrilateral where opposite sides are equal and parallel. This means the sides are reflected symmetrically across a diagonal axis.
In the context of complex numbers:

• If two opposite sides are equal, such as in the statement \(z_1 - z_4 = z_2 - z_3\), this implies the quadrilateral could be a parallelogram.
• The equal complex number differences (or vectors) indicate that the sides are parallel and of equal length.

The presence of these vectors suggests balance and symmetry, essential in recognizing the parallelogram nature of the shape.

Perpendicular Vectors

Vectors are mathematical objects with both magnitude and direction. In complex numbers, these are represented as differences between points (complex numbers).
When we say two vectors are perpendicular, it means they meet at a right angle. If vector \(z_4-z_1\) is perpendicular to \(z_2-z_1\), it follows that the angle between them is 90 degrees.
This concept, on its own, is crucial for determining shapes like rectangles.

• In the problem, the argument condition \(\text{amp}(\frac{z_4-z_1}{z_2-z_1})=\frac{\pi}{2}\) highlights the perpendicular nature of these vectors.
• This property, combined with equal opposite sides, confirms the quadrilateral is a rectangle (since rectangles have right angles).

Understanding perpendicular vectors helps in recognizing geometrical properties of shapes formed by complex numbers.