Problem 187 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: If the area of the triangle on the arganc plane formed by the complex numbers \(-z, i z, z-i z\) is 600 square units, then \(|z|=20\) Reason: Area of the triangle on the argand plane formed by the complex numbers \(-z, i z, z-i z\) is \(\frac{3}{2}\left|z^{2}\right|\)

Step-by-Step Solution

Answer

The modulus \\( |z| = 20 \\).

Step 1: Understand the Triangle Formulation

We are given three vertices of a triangle in the Argand plane: - Vertex 1 at \(-z\).- Vertex 2 at \(iz\).- Vertex 3 at \(z - iz\).

Step 2: Complex Number to Coordinates Conversion

Convert each complex number to corresponding coordinates:- \(-z\) is \((-x, -y)\).- \(iz\) is \((-y, x)\).- \(z - iz\) is \(x + y, y - x\).

Step 3: Use the Area of Triangle Formula

The formula for the area of a triangle formed by three points \( (x_1,y_1), (x_2,y_2), (x_3,y_3) \) is given by:\[\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]

Step 4: Substitute Coordinates into Area Formula

Substitute the coordinates of the points into the area formula:\[\text{Area} = \frac{1}{2} \left| -x(x-y) - y(y-(-x)) + (x+y)((-y)-x) \right| \]Simplify to get:\[= \frac{1}{2} \left| -2xy + y^2 - x^2 \right|\]

Step 5: Express \\( z \\) in terms of Modulus

Given that the area is 600 square units, equate and solve:\[\frac{1}{2} \left| -2xy + y^2 - x^2 \right| = 600\]The formula in given the reason states: \( \frac{3}{2} |z^2| \), equating both gives:\[\frac{3}{2} |z^2| = 600 \]

Step 6: Solve for \\( |z| \\)

From \[\frac{3}{2} |z^2| = 600\]solve for \(|z|\):\[|z^2| = 400 \]So, \(|z| = 20\).

Step 7: Conclusion

This confirms the assertion that if the area of the triangle is 600 square units, then the modulus of \(|z|\) must be 20, as both calculations match.

Key Concepts

Argand Plane

The concept of the Argand Plane is fundamental when dealing with complex numbers. It is a two-dimensional plane where each complex number is represented as a point. The horizontal axis is the real part, while the vertical axis is the imaginary part of a complex number. Hence, a complex number like \( z = x + yi \) gets a unique spot on this plane with coordinates \((x, y)\). The use of the Argand Plane allows us to transform complex number operations into geometric interpretations, making them easier to visualize and understand. This is particularly handy when considering transformations, rotations, or reflections accomplished by multiplying complex numbers. In the exercise, it describes three complex numbers used as vertices on this plane to form a triangle. Understanding how each complex number translates into a coordinate on the plane is crucial for calculating geometric properties such as areas.

Triangle Area

Understanding how to calculate the area of a triangle using vertices in a plane is a critical skill in geometry and is applied here in the Argand Plane. Given three points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the area \(T\) of the triangle can be calculated by the formula:

• \[ T = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \]

This formula is derived from the determinant of a matrix representing the coordinates, ensuring that the triangle's orientation doesn't affect the area calculation. In the context of the Argand Plane, substituting the coordinates derived from complex numbers into this formula allows us to compute the area without drawing or visualizing the triangle. The exercise used this formula to find an initial expression for the triangle's area in terms of the real and imaginary parts of the complex number \(z\). This process simplified the understanding of how geometric properties of complex numbers can be interpreted algebraically.

Modulus

The modulus of a complex number is analogous to the absolute value for real numbers. It represents the distance from the origin to the point \(z\) on the Argand Plane. For a complex number \(z = x + yi\), the modulus is calculated using:

• \[ |z| = \sqrt{x^2 + y^2} \]

This distance measure helps determine magnitudes in complex number applications, much like finding lengths in geometry. In this exercise, finding the modulus \(|z|\) was crucial for connecting the area of the triangle to a specific value, confirming the assertion about \(|z|\). The modulus simplifies the relationship between complex numbers and their geometric properties, allowing for clear insights into how transformations and distances manifest in the Argand Plane.