Problem 125 - A Complete Resource Book in Mathematics for JEE Main

Question

If in an argand plane points \(z_{1}, z_{2}, z_{3}\) are the vertices of an isosceles triangle right angled at \(z_{2}\), then (A) \(z_{1}^{2}+2 z_{2}^{2}+z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}\right)\) (B) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}\right)\) (C) \(z_{1}^{2}+z_{2}^{2}+2 z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}\right)\) (D) \(2 z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}\right)\)

Step-by-Step Solution

Answer

The correct choice is (A).

Step 1: Understand the Geometry of the Triangle

We have an isosceles right triangle in the Argand plane with vertices at points \(z_1, z_2, z_3\), and the right angle is at \(z_2\). This means the other two angles are \(45^\circ\). The distance from \(z_1\) to \(z_2\) is the same as the distance from \(z_3\) to \(z_2\). These distances can be represented as \(r\), where \(r = |z_2 - z_1| = |z_2 - z_3|\).

Step 2: Set Up Complex Rotation for Right Angle

Since the triangle is isosceles right, the line segment from \(z_1\) to \(z_2\) can be rotated by \(\frac{\pi}{2}\) to get the line segment from \(z_2\) to \(z_3\). This means \(z_3 = z_2 + i(z_1 - z_2)\). In the complex form, a rotation by \(90^\circ\) is represented by multiplication with \(i\).

Step 3: Derive Equation using Rotated Points

From \(z_3 = z_2 + i(z_1 - z_2)\), expand to get \(z_3 = z_2 + i z_1 - i z_2\). Simplifying, we get \(z_3 = (1-i)z_2 + i z_1\). Now, express \(z_1\), \(z_2\), and \(z_3\) in terms of this and substitute in equations to see what fits.

Step 4: Compare Options for Valid Equation

Substituting the expression for \(z_3\) into the given options and simplifying helps in checking which relation holds true. After trying the equations by substituting and simplifying, option (A) matches and satisfies the conditions.\(z_1^2 + 2z_2^2 + z_3^2 = 2z_2(z_1 + z_3)\) is verified.

Key Concepts

Argand Plane

The Argand plane is a way to represent complex numbers graphically. It's similar to an ordinary Cartesian coordinate system. Each complex number corresponds to a unique point in this plane.

• The horizontal axis (known as the real axis) represents the real part of a complex number.
• The vertical axis (known as the imaginary axis) represents the imaginary part of a complex number.

Understanding the Argand plane is crucial because it allows for geometric interpretations of complex number operations. In this context, you can visualize complex numbers as vectors.

For instance, the distance between two complex numbers in the Argand plane represents their magnitude difference. This is often pivotal when assessing shapes such as triangles, like the one in our exercise about the isosceles triangle.

Isosceles Triangle in Complex Plane

An isosceles triangle in the complex plane, especially one that is right-angled, gives us interesting geometrical insights.

• An isosceles triangle has two sides of equal length. If these sides are formed by complex numbers, their magnitudes are equal.
• In this specific problem, the right angle at point \(z_2\) implies that \(z_1\) and \(z_3\) are equidistant from \(z_2\). Hence, \(|z_2 - z_1| = |z_2 - z_3|\).

The property of isosceles triangles can be instrumental in forming equations that link vertices of the triangles. This equality of distances leads to utilizing algebraic relationships between the complex numbers to derive equations as shown in the step-by-step solution.
Knowing that the triangle has a right angle also leads to using trigonometric insights, like the 45-degree angles in this problem.

Complex Number Rotation

Complex number rotation is a powerful tool when dealing with geometric problems in the complex plane.

Rotation by a certain angle is achieved by multiplying a complex number by another complex number of unit magnitude (typically expressed as \(e^{i\theta}\) where \(\theta\) is the angle).
In this problem:

• Rotating by 90 degrees (or \(\frac{\pi}{2}\)) is equivalent to multiplying by \(i\).
• This operation modifies the coordinates in the Argand plane, effectively rotating the triangle around the origin or another vertex.

By applying this concept, the step-by-step solution uses the fact that you can express \(z_3\) in terms of \(z_1\) and \(z_2\) after applying a 90-degree rotation. Specifically, the rotation equations simplify to find expressions like \(z_3 = z_2 + i(z_1 - z_2)\), crucial in deducing the right relation and solving the initial problem.