Problem 150 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(O, A, B\) be three collinear points such that \(O A \cdot O B=1 .\) If \(O\) and \(B\) represent the complex numbers \(o\) and \(z\), then \(A\) represents (A) \(\frac{1}{\bar{z}}\) (B) \(\frac{1}{z}\) (C) \(\bar{z}\) (D) \(z^{2}\)

Step-by-Step Solution

Answer

A represents \( \frac{1}{\bar{z}} \).

Step 1: Understanding the Problem

We are given three collinear points \( O, A, B \) such that the product of distances \( OA \) and \( OB \) is 1. Points \( O \) and \( B \) are represented as complex numbers \( o \) and \( z \), respectively. We need to find the complex number representing the point \( A \).

Step 2: Distance in Complex Plane

In the complex plane, the distance between two points represented by complex numbers \( a \) and \( b \) is \(|a-b|\). Thus, \( OA = |a-o| \)}, \( OB = |z-o|\). The condition \( OA \cdot OB = 1 \) translates to \(|a-o| \cdot |z-o| = 1\).

Step 3: Using the Inverse Relation

If \(|a-o| \cdot |z-o| = 1\), then \( A \) is the inverse of \( B \) in relation to \( O \) in the complex plane. Therefore, the relation \( OA \cdot OB = 1 \) translates to \( a = \frac{1}{ar{z}} \), since \( ar{z} \) is the conjugate of \( z \) and gives the geometrical inverse across a unit circle centered at the origin.

Step 4: Conclusion

The point \( A \) is represented by the complex number that satisfies the inverse relation based on the given condition \( OA \cdot OB = 1 \). Therefore, the answer is \( \frac{1}{\bar{z}} \).

Key Concepts

Complex Plane

The complex plane, also known as the Argand plane, is a two-dimensional plane that graphically represents complex numbers. In this plane, each complex number corresponds to a unique point.
- The horizontal axis, known as the real axis, represents the real part of the complex number.
- The vertical axis, called the imaginary axis, represents the imaginary part.
For example, a complex number in the form of \( z = x + yi \) is plotted as the point \((x, y)\) on the complex plane, where \(x\) is the real part and \(y\) is the imaginary part.
This visualization allows us to utilize geometric concepts when analyzing complex numbers, making it easier to understand their properties and relationships.

Collinearity in Geometry

Collinearity is a fundamental concept in geometry where three or more points lie on a single straight line. In the context of the complex plane, if points are collinear, it means their corresponding complex numbers form a linear relationship.
For three points, \(O, A, B\), to be collinear:

• The vector from \(O\) to \(A\) must be a scalar multiple of the vector from \(O\) to \(B\).
• This can be represented as \(a - o = k(z - o)\) for some scalar \(k\).

In our given problem, we have further simplification where the condition \(OA \cdot OB = 1\) holds. This condition uses distances but also demands that the relationship between these points is mapped in the form of multiplication of distances equating to unity, suggesting a special symmetry among these points.

Geometrical Inversion

Geometrical inversion in complex analysis is a transformation that mirrors each point in the plane relative to a fixed circle. The most common scenario is inversion relative to the unit circle centered at the origin.
- The inverse of a complex number \(z\) with respect to the unit circle is given by the expression \( \frac{1}{ar{z}} \).
This transformation is significant because it maintains the basic structure of collinearity and distances when dealing with problems that require circular symmetry.
For instance, in our exercise, the condition \(OA \cdot OB = 1\) suggests an inversion relationship. That's because the distance product is unity, which corresponds precisely to the mapping property of geometric inversion relative to the unit circle. By this transformative property, point \(A\) is the inverse of point \(B\) in this special circular setting.