Problem 165 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The correct answer is (D): \(\frac{z_{1}-z_{2}}{z_{2}-z_{3}}\) is imaginary.

Step 1: Interpret the Problem

We have an isosceles right triangle with vertices at complex numbers \(z_1, z_2, z_3\), where the right angle is at \(z_2\). Our task is to determine which option correctly represents a relationship among \(z_1, z_2, z_3\).

Step 2: Establish Triangle Properties

For an isosceles right triangle, if it is right-angled at \(z_2\), two sides (\(z_1z_2\) and \(z_2z_3\)) must be equal in length and the angle between these sides is 90 degrees. This implies that the difference between the points representing these sides in the complex plane should satisfy the Pythagorean theorem.

Step 3: Convert to Algebraic Form

Express the distances between the points using complex number subtraction: \(|z_1 - z_2| = |z_2 - z_3|\) and the Pythagorean relation: \(|z_1 - z_3|^2 = |z_1 - z_2|^2 + |z_2 - z_3|^2\).

Step 4: Evaluate Given Options

We check each equation option to determine if it reflects these properties:- (A) Resembles a quadratic Pythagorean identity.- (B) Matches the Pythagorean theorem if rewritten as \((z_1 - z_2)^2 + (z_3 - z_2)^2 = |z_1 - z_3|^2\).- (C) Would imply a contradiction since it means the triangle degenerates.- (D) Reflects a relationship in terms of complex numbers.

Step 5: Analyze Option D

The ratio \(\frac{z_1 - z_2}{z_2 - z_3}\) should be imaginary if the line segments \(z_1z_2\) and \(z_2z_3\) form a right angle. For a right-angled triangle, this ratio being imaginary indicates the perpendicularity property from vectors.

Key Concepts

Right-Angle Triangle

A right-angle triangle is a special type of triangle where one of its angles measures exactly 90 degrees. This angle forms a perpendicular intersection between two of the triangle's sides, known as the legs. The side opposite the right angle is called the hypotenuse and is typically the longest side. In a right-angle triangle, the Pythagorean theorem applies, which states that the square of the hypotenuse (\(c\)) equals the sum of the squares of the other two sides (\(a\) and \(b\)).

• Mathematically, it's represented as: \(c^2 = a^2 + b^2\).
• This theorem helps to compute distances and verify angles.

Isosceles right-angle triangles have special properties, as two sides are not only equal but the angle opposite one of these equal sides is a right angle. This means the other two angles must both be 45 degrees. This unique setup indicates that both legs are congruent, adding symmetry to the shape and simplifying calculations.

Complex Numbers

Complex numbers extend the idea of a number beyond the real number line, incorporating imaginary units. A complex number is of the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) representing the square root of \(-1\). These numbers are often used in two-dimensional space as points or vectors, making them incredibly useful in representing geometric transformations and other spatial concepts.

• They allow calculations that cannot be solved using only real numbers.
• Complex numbers can effectively illustrate transformations and rotations.

In the context of isosceles right triangles, placing vertices as complex numbers allows us to calculate distances using the modulus, which acts like a magnitude or distance measurement in the complex plane. It simplifies understanding rotations or transformations that correspond to spatial movements, aligning perfectly with triangle calculations involving angles and side lengths.

Triangle Properties

Triangles, by definition, are three-sided polygons with three vertices and three angles, the sum of which is always 180 degrees. A triangle’s properties significantly affect geometric comprehension and calculations. Notably, different types include

• Equilateral – three equal sides and three equal angles.
• Isosceles – two equal sides, two equal angles.
• Scalene – all sides and angles are different.

The triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third, is crucial in validating the existence of a triangle. Particularly for isosceles right-angle triangles, recognizing congruent sides provides simplification in problem-solving. Such triangles often appear in complex number problems due to their symmetric nature, which reduces the complexity of calculation and analysis.