Problem 134 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1}, z_{2}, z_{3}\) are non-zero, non-collinear complex numbers such that \(\frac{2}{z_{1}}=\frac{1}{z_{2}}+\frac{1}{z_{3}}\), then the points \(z_{1}, z_{2}, z_{3}\) lie (A) in the interior of a circle (B) on a circle passing through origin (C) in the exterior of a circle (D) None of these

Step-by-Step Solution

Answer

(B) on a circle passing through origin

Step 1: Understanding the Equation

We are given \( \frac{2}{z_{1}} = \frac{1}{z_{2}} + \frac{1}{z_{3}} \). This can be rewritten as \( \frac{2}{z_{1}} - \frac{1}{z_{2}} - \frac{1}{z_{3}} = 0 \). By finding a common denominator, this equation becomes \( \frac{2z_2z_3 - z_1z_3 - z_1z_2}{z_1z_2z_3} = 0 \), which simplifies to \( 2z_2z_3 = z_1z_3 + z_1z_2 \).

Step 2: Reorganizing Terms

Let's rearrange the equation \(2z_2z_3 = z_1z_3 + z_1z_2\) as \( 2z_2z_3 - z_1z_3 - z_1z_2 = 0 \) or \( z_2(2z_3 - z_1) - z_1z_3 = 0 \). Factor out \(z_3\) to get \( (2z_2 - z_1)z_3 = z_1z_2 \).

Step 3: Analyzing Vector Representation

To interpret this geometrically, consider representing these complex numbers as vectors in the complex plane. If we take \(z_1, z_2, z_3\) as vector endpoints, the equation becomes relevant to Apollonius Circle Theorem in geometry, which shows the ratios of distances from a point to two fixed points form a circle.

Step 4: Interpreting the Geometric Condition

The condition \( \frac{2}{z_1} = \frac{1}{z_2} + \frac{1}{z_3} \) implies a harmonic division. In the complex plane, this means the points \( z_1, z_2, \) and \( z_3 \) lie on an Apollonius circle of which the origin is on the circle.

Key Concepts

Apollonius Circle Theorem

The Apollonius Circle Theorem is a fascinating concept in geometry that relates to the position of points in a plane. It's named after the Greek mathematician Apollonius of Perga. This theorem states that, for any given point, there is a circle such that the distance of any point on this circle from two fixed points is in a constant ratio.
To put it simply, imagine you have two fixed points, say A and B, and you pick a point C somewhere on the plane. The Apollonius circle would include all points, like C, where the distance to A and B maintains a specific ratio. In the context of complex numbers, this can translate into equations involving those numbers.

• The Apollonius circle offers insight into how these complex numbers relate to each other spatially.
• It provides a geometric interpretation, helping us visualize equations as circular paths.

When interpreting complex equations like \( \frac{2}{z_1} = \frac{1}{z_2} + \frac{1}{z_3} \), it indicates that points \(z_1, z_2, \) and \(z_3\) lie on such an Apollonius circle, revealing proportional distances within a geometric circle centered around the origin.

Vector Representation in Complex Plane

The vector representation in the complex plane is a powerful way to visualize and solve equations involving complex numbers. Each complex number can be seen as a vector originating from the origin of the plane, with its endpoint at the coordinates defined by the real and imaginary parts.
In simple terms, it's like viewing each complex number as an arrow that points somewhere on the plane. This arrow's length and direction are determined by the number's magnitude and argument.

• Vector representation helps to uncover the spatial relationships among complex numbers.
• It allows us to see geometric patterns, like those formed by circles or lines connected by these vectors.

For the equation \( \frac{2}{z_1} = \frac{1}{z_2} + \frac{1}{z_3} \), visualizing each of these \(z\) values as vectors helps grasp how changes in direction and magnitude affect their interaction. The interplay of these vectors essentially leads us to explore patterns that align with the Apollonius Circle Theorem.

Harmonic Division in Complex Plane

Harmonic division is an intriguing concept, particularly in the field of projective geometry, which extends into the complex plane. In essence, a harmonic division is a partitioning of a line segment into four points \(A, B, C, D\) such that the cross-ratio \( (A, B; C, D) \) equals -1. This special condition relates the perpendicular distances or vectors stemming from these points, even if they are complex values.
In the equation \( \frac{2}{z_1} = \frac{1}{z_2} + \frac{1}{z_3} \), the concept of harmonic division aids in understanding how these points are positioned to form a balanced or "harmonic" layout around a given circle or line.

• Harmonic division in the complex plane highlights a specific symmetry and division of areas.
• This adds depth to interpreting equations involving more than one point or line segment.

By visualizing how \(z_1, z_2, \) and \(z_3\) divide the plane harmonically, we better understand their geometric placement and interaction, particularly in relation to circular formations like those described by Apollonius circles.