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

Question

The triangle formed by the points \(1, \frac{1+i}{\sqrt{2}}\) and \(i\) as 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: Plot the points on the complex plane

First, identify the points on the Argand diagram: the points are represented by the complex numbers 1, \(\frac{1+i}{\sqrt{2}}\), and \(i\). The point 1 corresponds to (1,0), \(\frac{1+i}{\sqrt{2}}\) simplifies to \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\), and \(i\) corresponds to (0,1).

Step 2: Calculate the distances between the points

Find the length of each side of the triangle formed by these points: - Distance between \(1\) and \(\frac{1+i}{\sqrt{2}}\): \[\sqrt{\left(1 - \frac{1}{\sqrt{2}}\right)^2 + \left(0 - \frac{1}{\sqrt{2}}\right)^2} = \sqrt{2 - \sqrt{2}}\]- Distance between \(\frac{1+i}{\sqrt{2}}\) and \(i\): \[\sqrt{\left(\frac{1}{\sqrt{2}} - 0\right)^2 + \left(\frac{1}{\sqrt{2}} - 1\right)^2} = \sqrt{2 - \sqrt{2}}\]- Distance between \(1\) and \(i\): \[\sqrt{(1-0)^2 + (0-1)^2} = \sqrt{2}\]

Step 3: Classify the triangle based on the side lengths

Compare the distances calculated: The two sides between \(1\) and \(\frac{1+i}{\sqrt{2}}\), and \(\frac{1+i}{\sqrt{2}}\) and \(i\) are equal. The side between \(1\) and \(i\) is different. Hence, two sides are equal, making the triangle isosceles.

Key Concepts

Argand Diagram

The Argand Diagram is a way to visualize complex numbers on a plane. Just like a traditional Cartesian coordinate system, this diagram has two axes:

• The horizontal axis represents the real part of a complex number.
• The vertical axis represents the imaginary part.

To plot a complex number like \(a + bi\), you locate \(a\) on the real axis and \(b\) on the imaginary axis. For example, using the numbers in the exercise, the number \(1\) corresponds to the point \((1,0)\), the point \(i\) is \((0,1)\), and \(\frac{1+i}{\sqrt{2}}\) plots to \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\).
The Argand Diagram helps us visualize relationships like distances and angles between complex numbers, which is crucial for interpreting their geometrical properties.

Distance Formula

The distance formula is a tool derived from the Pythagorean theorem, helping us find the distance between two points in a plane. When considering complex numbers on an Argand Diagram, points are simply the graphical representation of these numbers.

• To compute the distance between two complex numbers \(a+bi\) and \(c+di\), use the formula: \(\sqrt{(a-c)^2 + (b-d)^2}\).

For example, consider calculating the distance from \(1\) to \(\frac{1+i}{\sqrt{2}}\):
The distance is \((\sqrt{(1 - \frac{1}{\sqrt{2}})^2 + (0 - \frac{1}{\sqrt{2}})^2})\), which simplifies to \(\sqrt{2 - \sqrt{2}}\).
This formula assists in identifying the lengths of each side of a triangle formed on the complex plane.

Properties of Triangles

Triangles have certain properties that can be determined by their side lengths and angles. In this exercise, we focus on classifying a triangle formed by complex numbers on an Argand Diagram by its sides.
An isosceles triangle has two sides of equal length, while a scalene triangle has all sides different and an equilateral triangle has all sides equal.

• The distance calculations, as seen in the solution, help verify the side lengths:
• The two sides \(1\) to \(\frac{1+i}{\sqrt{2}}\) and \(\frac{1+i}{\sqrt{2}}\) to \(i\) are equal.
• The side \(1\) to \(i\) is different. Thus, the triangle is isosceles because two sides are of equal length.

Understanding these properties helps in determining the type of triangle simply by calculating distances with the Argand Diagram.