Problem 165 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}, z_{2}, z_{3}\) are the vertices of an isosceles triangle and right angled at \(z_{2}\), then (A) \(z_{1}^{2}+z_{3}^{2}+2 z_{2}^{2}=2\left(z_{1}+z_{3}\right) z_{2}\) (B) \(z_{1}^{2}+z_{3}^{2}=2 z_{2}\left(z_{1}+z_{3}-z_{2}\right)\) (C) \(\left(z_{1}-z_{2}\right)^{2}+\left(z_{2}-z_{3}\right)^{2}=0\) (D) \(\frac{z_{1}-z_{2}}{z_{2}-z_{3}}\) is imaginary

Step-by-Step Solution

Answer

Option (D) is correct: \(\frac{z_1-z_2}{z_2-z_3}\) is imaginary.

Step 1: Identify the Triangle Type and Vertices

Given that the triangle is isosceles and right-angled at \(z_2\), it means two sides are equal and one angle is 90 degrees. Let's assume the hypotenuse is between \(z_1\) and \(z_3\), and \(z_2\) is the right angle.

Step 2: Apply the Perpendicularity Condition Using Complex Numbers

For a right angle at \(z_2\), vectors \(z_1 - z_2\) and \(z_3 - z_2\) must be perpendicular. In terms of complex numbers, the dot product condition translates to the imaginary, i.e., \((z_1 - z_2) / (z_2 - z_3)\) being imaginary. This matches option (D).

Step 3: Conclude the Valid Expression for the Triangle

The condition derived from complex number representation verifies that the expression \(\frac{z_1-z_2}{z_2-z_3}\) is indeed imaginary, which aligns with choice (D). The other options do not fulfill the condition for perpendicular vectors.

Key Concepts

Complex Plane Geometry

In mathematics, the concept of complex plane geometry is a powerful tool to study the properties of shapes and figures using complex numbers. Every complex number can be represented as a point on a complex plane, which is a two-dimensional plane consisting of a real axis and an imaginary axis.
The coordinates of these points correspond to the real part and imaginary part of the complex number. Thus, any complex number, say \(z = a + bi\), can be perceived as a point with coordinates \((a, b)\) on this plane.
Complex plane geometry allows for the visualization of shapes such as triangles and circles using complex numbers. It simplifies the study of these geometries because operations like addition, subtraction, and finding perpendiculars translate to simple algebraic operations on complex numbers.
In problems involving complex geometry, like in triangles defined by vertices \(z_1, z_2, z_3\), it’s important to understand how the position of these points and their relationships can be expressed and manipulated using complex arithmetic.

Right-Angled Triangles

Right-angled triangles are a special category of triangles where one of their angles is exactly 90 degrees. In the case of our triangle with vertices \(z_1, z_2, z_3\), the right angle is at \(z_2\).
This implies that the sides adjacent to \(z_2\) form perpendicular vectors. The concept of perpendicularity is fundamental here because it helps determine the specific relationships between the sides of the triangle.
In terms of complex numbers, the sides of the triangle can be represented as vectors in the plane. For right-angled triangles, the magnitude of the hypotenuse (the side opposite the right angle) can be determined using the Pythagorean theorem, where the sum of the squares of the lengths of the other two sides equals the square of the hypotenuse.

Isosceles Triangles

An isosceles triangle is characterized by having exactly two sides of equal length. In the context of our problem, the triangle formed by \(z_1, z_2, z_3\) is isosceles, which suggests symmetry in the triangle’s geometry.
This symmetry simplifies various calculations. When dealing with such triangles on the complex plane, understanding which sides are equal helps establish relations among the complex numbers representing the vertices of the triangle.
For example, if \(z_2\) is the right angle and the sides \(z_1z_2\) and \(z_2z_3\) are equal, it implies a certain predictability and regularity in the problem structure. This can be leveraged to apply mathematical conditions or transformations using complex numbers to deduce specific properties or solve for unknowns.

Perpendicular Vectors in Complex Form

In complex numbers, perpendicular vectors play a crucial role, especially when studying triangles. When two vectors are perpendicular, their dot product is zero. In complex form, this means that the quotient of two vectors is purely imaginary.
For our triangle, with a right angle at \(z_2\), the vectors \(z_1 - z_2\) and \(z_3 - z_2\) must be perpendicular. This condition is mathematically represented as \(\frac{z_1 - z_2}{z_3 - z_2}\) being purely imaginary. This confirms that one complex vector is a rotation of the other by 90 degrees.
This perpendicularity is critical for solving problems involving right-angled triangles using complex numbers, as it dictates certain algebraic properties of the numbers involved and indicates specific relationships among them in the complex plane.