Problem 121 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(|z-25 i| \leq 15\), then |maximum amp \((z)-\) minimum \(\operatorname{amp}(z) \mid\) is equal to (A) \(\sin ^{-1}\left(\frac{3}{5}\right)-\cos ^{-1}\left(\frac{3}{5}\right)\) (B) \(\frac{\pi}{2}+\cos ^{-1}\left(\frac{3}{5}\right)\) (C) \(\pi-2 \cos ^{-1}\left(\frac{3}{5}\right)\) (D) \(\cos ^{-1}\left(\frac{3}{5}\right)\)

Step-by-Step Solution

Answer

The correct answer is (C): \(\pi - 2 \cos^{-1}\left(\frac{3}{5}\right)\).

Step 1: Interpret the Complex Inequality

The expression \(|z - 25i| \leq 15\) describes a region in the complex plane. Specifically, it represents a closed disk centered at the point \((0, 25)\) (or simply \(0 + 25i\) in complex terms) with a radius of 15.

Step 2: Visualize the Region

In the complex plane, visualize a circle centered at \((0, 25)\) with a radius of 15. This circle includes all the points \(z = x + yi\) that are within or on the boundary up to a distance of 15 from the center.

Step 3: Find the Range of Argument for z

The argument of a complex number \(z = x + yi\) is the angle \(\theta\) between the positive real axis and the line connecting \(z\) with the origin. Hence, we need to find the maximum and minimum angles \(\theta\) that correspond to points on the boundary of the circle.

Step 4: Use Geometry to Determine Critical Angles

The vertical distance from the center \((0, 25)\) to the x-axis is 25. The radius of the circle is 15, which makes the tangents from the origin to the circle form right triangles. Using trigonometry, these tangent lines will form angles whose sine or cosine will relate to \(\frac{3}{5}\).

Step 5: Calculate Maximum and Minimum Arguments

The maximum amplitude occurs when \(z\) is at the top of the circle, and minimum amplitude occurs at the bottom. The key angles form arcs defined by \(\sin^{-1}\left(\frac{3}{5}\right)\) and \(\cos^{-1}\left(\frac{3}{5}\right)\).

Step 6: Evaluate the Given Choices

For maximum and minimum angle difference, \(\pi - 2 \cos^{-1}\left(\frac{3}{5}\right)\) corresponds to the range covering the circle's topmost to bottommost points from the perspective of origin.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane where every point represents a complex number. Each complex number can be written in the form \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part. In this plane:

• The horizontal axis (x-axis) represents the real component.
• The vertical axis (y-axis) represents the imaginary component.

For instance, the inequality \(|z-25i| \leq 15\) describes a closed region, specifically a circle, on this complex plane. The center of this circle is at \((0, 25)\), and it includes all complex numbers \(z = x + yi\) that are no more than 15 units away from this center point.
Understanding the complex plane allows us to visualize complex numbers much like vectors in a plane, where magnitude and direction are crucial aspects to consider.

Trigonometry

Trigonometry is a branch of mathematics focused on the relationships between angles and lengths in triangles. In the context of complex numbers:

• The argument (or amplitude) of a complex number is the angle \(\theta\) that it makes with the positive real axis.
• This angle can be determined using the tangent function as \(\tan\theta = \frac{y}{x}\), where \(x\) and \(y\) are the real and imaginary parts, respectively.

In problems involving inequalities like \(|z-25i| \leq 15\), we often use trigonometric identities to find critical angles. For instance, right triangles formed by tangents to a circle in the complex plane assist in finding sine and cosine relationships, such as \(\sin^{-1}(\frac{3}{5})\) and \(\cos^{-1}(\frac{3}{5})\). Understanding these trigonometric identities helps in calculating the range of possible angles that describe the positions of complex numbers with respect to a given point.

Geometric Interpretation of Inequalities

Interpreting inequalities geometrically involves visualizing the sets of points that satisfy the inequality. In the context of complex numbers:

• The expression \(|z-25i| \leq 15\) signifies all points within or on the boundary of a circle.
• This circle is centered at the imaginary component, \(25i\), and has a radius of 15 units.

Geometrically, the maximum and minimum arguments of complex numbers on this circle indicate the largest and smallest angles that can be formed with the origin. These are found at the topmost and bottommost points of the circle when visualized from the origin. The solution \(\pi - 2 \cos^{-1}(\frac{3}{5})\) represents the angular coverage from the top to the bottom of the circle. This geometric understanding translates into real-world interpretations essential for solving and visualizing complex number properties by their spatial relationships.