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

Question

Assertion: If the area of the triangle on the argand 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

Assertion is correct; \(|z| = 20\).

Step 1: Understand the Complex Points

The vertices of our triangle in the Argand plane are given by complex numbers: - \(-z\)- \(iz\)- \(z - iz\)These represent points \((-a, -b)\), \((-b, a)\), and \((a-b, b+a)\) on the Argand diagram, assuming \(z = a + bi\).

Step 2: Apply the Formula for Area

The reason states that the area \(A\) of the triangle formed by these points in the Argand plane is given by:\[ A = \frac{3}{2}|z^2| \]where \(|z|\) is the magnitude of the complex number \(z\). The given area is 600 square units.

Step 3: Calculate Magnitude from Area

We know the area is 600, so we set up the equation:\[ \frac{3}{2}|z^2| = 600 \]To solve for \(|z|^2\), multiply both sides by \(\frac{2}{3}\):\[ |z^2| = 400 \]

Step 4: Solve for |z|

Since \(|z^2| = |z|^2\):\[ |z|^2 = 400 \]Taking the square root of both sides yields:\[ |z| = \sqrt{400} = 20 \]

Step 5: Verification of Assertion

The calculated magnitude \(|z|\) is indeed 20, which matches the assertion that \(|z| = 20\) if the area of the triangle is 600.

Key Concepts

Argand Plane

The Argand plane is a powerful tool used to visualize complex numbers as points in a two-dimensional plane. Each complex number, represented by \(z = a + bi\), corresponds to a point in this plane. The horizontal axis, known as the real axis, represents the real part \(a\), while the vertical axis, the imaginary axis, represents the imaginary part \(b\).

When we handle complex numbers graphically, it becomes easier to interpret relationships and transformations. For instance, the complex numbers \(-z\), \(iz\), and \(z - iz\) translate to specific coordinates within this plane. These points can act as vertices of geometrical shapes like triangles. Understanding this spatial representation simplifies complex number operations and gives them a visual context.

Triangle Area

To determine the area of a triangle formed by complex numbers, we use certain geometric principles. In the Argand plane, the vertices of a triangle can be represented by complex numbers like \(-z\), \(iz\), and \(z - iz\). Given these points, one formula to compute the area of a triangle is:

• Area \(A = \frac{3}{2}|z^2|\)

This formula leverages the properties of magnitudes in complex number theory. Essentially, it simplifies calculations by utilizing the squared magnitude \(|z^2|\), which directly affects the size of the triangle in the Argand plane.

When given a triangle's area, you can reverse-engineer the problem to find characteristics of the complex numbers involved, such as their magnitude \(|z|\), which is crucial for understanding the scale and scope of the triangle in question.

Magnitude of Complex Numbers

The magnitude of a complex number, often denoted as \(|z|\), provides a measure of its distance from the origin in the Argand plane. For a complex number \(z = a + bi\), its magnitude is calculated using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

This is analogous to finding the length of the hypotenuse of a right-angled triangle with sides \(a\) and \(b\).

In the context of the original exercise, knowing the magnitude \(|z|\) is key to resolving the problem. If the area of a triangle is known, it can be used to find the square of the magnitude through the relationship \(|z|^2 = \frac{2}{3}A\), and vice versa. Magnitude serves as a bridge between abstract algebraic expressions and their tangible geometric interpretations.