Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The triangle formed by the points \(1, \frac{1+i}{\sqrt{2}}\) and \(i\) vertices in the Argand diagram is (A) scalene (B) equilateral (C) isosceles (D) right-angled

Step-by-Step Solution

Answer

The triangle is isosceles.

Step 1: Identify the vertices in the Complex Plane

We are given three points in the complex plane: \(1\), \( \frac{1+i}{\sqrt{2}} \) and \( i \). These represent vertices of a triangle on the Argand diagram.

Step 2: Convert Complex Numbers to Cartesian Coordinates

The complex number \(1\) is at point \((1, 0)\), \(\frac{1+i}{\sqrt{2}}\) is at point \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\), and \(i\) is at point \((0, 1)\) in the Argand plane.

Step 3: Calculate the Lengths of the Sides

Use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]- For points \((1, 0)\) and \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\): \[ \sqrt{\left(\frac{1}{\sqrt{2}} - 1\right)^2 + \left(\frac{1}{\sqrt{2}} - 0\right)^2} = \sqrt{2 - \sqrt{2}} \]- For points \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\) and \((0, 1)\): \[ \sqrt{\left(0 - \frac{1}{\sqrt{2}}\right)^2 + \left(1 - \frac{1}{\sqrt{2}}\right)^2} = \sqrt{2 - \sqrt{2}} \]- For points \((1, 0)\) and \((0, 1)\): \[ \sqrt{(0 - 1)^2 + (1 - 0)^2} = \sqrt{2} \]

Step 4: Analyze the Lengths of the Sides

We have determined that two sides of the triangle have the same length: \(\sqrt{2 - \sqrt{2}}\), and one side is different, with length \(\sqrt{2}\). Thus, the triangle is isosceles, as it has at least two sides of equal length.

Key Concepts

Argand Diagram

Imagine a graph where we can represent complex numbers using both a real and an imaginary axis. This is known as the Argand Diagram. It is like a map where each point describes a specific location based on a real part and an imaginary part.
For example, the number represented by a complex coordinate like \(1 + i\), translates to a point on this diagram. The real part, 1, tells you how far to move along the horizontal (real) axis. The imaginary part, \(i\), tells you how far to move along the vertical (imaginary) axis.

• The point \((1, 0)\) corresponds to the number 1.
• The point \((0, 1)\) corresponds to the number \(i\).
• The point \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\) corresponds to \(\frac{1+i}{\sqrt{2}}\).

Understanding the Argand Diagram is like knowing how to read a map. It helps visualize complex numbers in a way that's easy to understand.

Distance Formula

The distance formula is a simple tool to measure how far apart two points are on a plane. It is derived from the Pythagorean theorem, stating that:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Let's break it down. Every point in the diagram can be thought of like coordinates on a grid. To find how far one point is from another, you calculate sideways (x-axis) and upwards (y-axis) differences and use them in this formula.

• This formula is crucial for determining the lengths of the sides in triangles.
• When applied to the points provided, it showed that two sides of the triangle have the same length.

This indicates that you don't need fancy tools to measure distance, just a little understanding of how to use this formula.

Isosceles Triangle

An isosceles triangle is special because it has at least two sides that are of equal length. This unique characteristic not only makes it a key shape in geometry, but also simplifies calculations in many math problems.

• For our exercise, the triangle formed on the Argand Diagram has two sides of equal length.
• These sides were confirmed using the distance formula.

This tells us that even on a complex number plane, the properties of triangles are still dependable. Recognizing the isosceles nature of a triangle can help us solve complex problems more easily.

Complex Plane

The complex plane is a two-dimensional plane similar to the Cartesian plane you use for regular numbers but with a twist — it’s specifically for complex numbers. Here each number is represented by a point, where the horizontal line is the real part, and the vertical line is the imaginary part.

• Think about the complex number \(a + bi\). "a" is the horizontal coordinate and "b" is the vertical one.
• Every complex number appears as one unique point on this plane.

Understanding the complex plane is essential for visualizing and solving problems involving complex numbers, as it turns what could be abstract into something we can easily see and understand.

Cartesian Coordinates

Cartesian coordinates provide a straightforward way of representing points in a plane using two axes: an x-axis (horizontal) and a y-axis (vertical). This system is foundational not only in mathematics but also in fields like physics and engineering.

In our exercise, complex numbers were converted into Cartesian coordinates, which aided in visualizing them as points on the Argand Diagram.

• This conversion is key for finding distances between points using the distance formula.
• Thanks to these coordinates, identifying whether triangles are isosceles becomes simpler.

A firm grasp of Cartesian coordinates can make solving mathematical problems a walk in the park. They transform abstract numbers into tangible points for easy comprehension and manipulation.