Problem 149 - A Complete Resource Book in Mathematics for JEE Main

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

The answer is (A) \(\frac{1}{\bar{z}}\).

Step 1: Understand the Problem

We are given that the points \(O, A, B\) are collinear, and the product of the distances \(OA \cdot OB = 1\). We also know that \(O\), \(B\), and \(A\) represent complex numbers \(o\), \(z\), and some unknown \(a\) respectively. We need to find the expression for \(a\).

Step 2: Write the Collinear Condition in Terms of Complex Numbers

Since \(O, A, B\) are collinear on the complex plane, the points can be represented as complex numbers. Let \(O = o\), \(B = z\), \(A = a\). Given that \(OA \cdot OB = 1\), we express this condition in terms of complex numbers: \(|a - o| \cdot |z - o| = 1\).

Step 3: Apply the Value of O as Origin

Assuming \(o = 0\) (since \(O\) is often the origin), the expression simplifies to \(|a| \cdot |z| = 1\). This implies that the magnitudes of \(a\) and \(z\) multiplied together give 1.

Step 4: Solve for A in Terms of Z

From \(|a| \cdot |z| = 1\), we find \(|a| = \frac{1}{|z|}\). Since \(a\) needs to be determined in terms of \(z\), we express \(a\) as \(a = \frac{1}{ar{z}}\), because multiplying by the conjugate maintains the magnitude condition \(|z|^2 = z \cdot \bar{z}\).

Step 5: Identify the Correct Option

We determined earlier that \(a = \frac{1}{\bar{z}}\). Comparing it with the given options, option (A) is the one that matches our result: \(\frac{1}{\bar{z}}\).

Key Concepts

Collinearity

Collinearity in geometry refers to the arrangement where three or more points lie on the exact same line. In the context of complex numbers and the complex plane, collinearity allows us to analyze these points in terms of their complex representations. This geometric property implies that if points are collinear, any point between the other two can be described as a linear combination of the other points.

For example, if points \(O, A, B\) are collinear, there exists a real number \(t\) such that \(A = O + t(B - O)\). When translated into complex numbers, collinearity can yield valuable equations used for solving problems, such as those involving distances or magnitudes, as seen in this exercise. This becomes especially useful when tackling problems in the complex plane, where visual intuition might be less straightforward than simple arithmetic.

Understanding collinearity helps simplify problems in fields like physics and engineering, where points may need to be aligned with certain axes or directions to simplify calculations.

Complex Plane

The complex plane is a two-dimensional plane used to visualize and perform operations on complex numbers. Complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with \(i^2 = -1\).

In this plane, the horizontal axis (also known as the real axis) represents the real part of a complex number, while the vertical axis (imaginary axis) represents the imaginary part. This makes every point on the plane uniquely defined by its complex coordinate.

• Using the complex plane, rotational transformations and scaling can be elegantly represented as complex multiplication.
• Addition of complex numbers corresponds to vector addition in this plane.

By plotting points that correspond to complex numbers, one can easily understand relationships between values, such as collinearity, distance, and magnitudes, which can be pivotal for analyzing transformations and relations in advanced mathematics or physics. The complex plane serves as a bridge, providing geometric interpretation to otherwise abstract algebraic operations.

Magnitudes of Complex Numbers

The magnitude of a complex number, also known as its modulus, measures its size or distance from the origin in the complex plane. It is denoted by \(|z|\) for a complex number \(z = a + bi\) and calculated using the formula \(|z| = \sqrt{a^2 + b^2}\).

The magnitude reflects the length of the vector representing the complex number, making it crucial for understanding the geometric representation in the complex plane. This concept is extensively used in problems dealing with distances between complex numbers or in establishing certain multiplicative properties, such as those seen where magnitudes must satisfy specific conditions, like in the collinearity exercise given.

• Multiplying a complex number by its conjugate results in a real number, precisely the square of its magnitude: \(z\bar{z} = |z|^2\).
• This property is particularly useful in simplifying expressions and solving equations involving complex numbers.

Understanding magnitudes thus empowers students to tackle complex problems with confidence, giving them tools to evaluate both the position and relationship of points in the complex plane through straightforward calculations.