Problem 135 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(|z-25 i| \leq 15\), then the least positive value of arg \(z\) is \(\begin{array}{ll}\text { (A) } \pi-\tan ^{-1} \frac{4}{3} & \text { (B) } \tan ^{-1} \frac{4}{3}\end{array}\) (C) \(-\pi+\tan ^{-1} \frac{4}{3}\) (D) None of these

Step-by-Step Solution

Answer

The least positive value of \( \arg(z) \) is \( \tan^{-1} \frac{4}{3} \), corresponding to option (B).

Step 1: Understanding the Problem

We need to find the minimum positive value of the argument of \( z \) given the condition \( |z - 25i| \leq 15 \). This inequality represents a disk (circle region) in the complex plane with center at \( 25i \) and radius 15.

Step 2: Expressing z in Coordinate Form

Express \( z \) in terms of real and imaginary parts, i.e., \( z = x + yi \). The condition can be rewritten as \( |x + (y - 25)i| \leq 15 \).

Step 3: Translating the Inequality

Translate the inequality to coordinates: \( \sqrt{x^2 + (y-25)^2} \leq 15 \). This is a circle centered at (0, 25) with a radius of 15 in the complex plane.

Step 4: Analyzing Points on the Boundary

Consider the boundary where \( \sqrt{x^2 + (y-25)^2} = 15 \). We want the smallest positive argument, which typically occurs along the positive x-axis.

Step 5: Finding the Point with Minimum Argument

Understand that the smallest positive argument is at the point on the circle with the largest positive x-value and minimum y-value on the boundary. This corresponds to the angle made by the line from the origin to this point.

Step 6: Calculating the Argument

The least positive argument occurs at \((x, y) = (12, 20)\) since this point has the largest positive real component (x) and fits within the circle. Calculate \( \tan^{-1}(20/12) = \tan^{-1}(5/3) \).

Step 7: Checking Answer Against Options

Compare the calculated argument to the provided options. Check which answer corresponds to \( \tan^{-1} \frac{4}{3} \) understandably given as an approximation.

Key Concepts

Argument of a Complex Number

When dealing with complex numbers, each one can be represented in a plane with a real part and an imaginary part. The *argument* of a complex number is an angle that tells us the direction in which the number points from the origin. This angle is measured in radians and goes counterclockwise from the positive real axis.
Let's consider a complex number like \( z = x + yi \). The argument of this complex number, denoted as \( \arg(z) \), is calculated using the arctangent function:

• \( \arg(z) = \tan^{-1}\left(\frac{y}{x}\right) \)

Remember that the argument is usually given in a range from \(-\pi\) to \(\pi\). The smallest positive argument is crucial in understanding the orientation of points in the context of this exercise. Here, we must locate the point which maximizes the x-value while conforming to a given constraint. This constraint helps us find the smallest positive argument, guiding us towards the correct solution.

Geometric Representation

The geometric representation is essential in visualizing complex numbers on a plane, known as the complex plane. In this scenario, complex numbers are treated as vectors originating from the origin. The real component \( x \) is plotted along the horizontal axis (x-axis), and the imaginary component \( y \) is plotted along the vertical axis (y-axis).
In the problem's context, a geometric figure in the shape of a circle (or disk) is formed using the inequality \(|z - 25i| \leq 15\). This tells us that all points \( z \) that satisfy this condition lie within or on a circle centered at the point \((0,25)\) with a radius of 15.

• The center of the circle is provided by the complex number's specificity in the inequality, \( 25i \), which implicitly gives the center as \((0, 25)\) in the coordinates.
• This visualization helps in understanding which part of the plane is relevant and assists in determining key points such as the smallest positive argument.

Inequality in Complex Plane

Inequalities involving complex numbers often define regions within the complex plane, like disks or circles. Here, the inequality \(|z - 25i| \leq 15\) expresses such a region.
Breaking it down:

• The expression \(|z - 25i|\) represents the distance between a point \( z \) (with real part \( x \) and imaginary part \( y \)) and the point \( 25i \) on the complex plane.
• Using Pythagoras' theorem, this distance is given by \(\sqrt{x^2 + (y-25)^2}\).
• The inequality \(\leq 15\) sets a boundary for this distance; points lying at a distance less than or equal to 15 from \((0, 25)\) are encompassed by the circle.

Understanding these distances and their boundaries is key to solving problems regarding certain constraints in the complex plane. It directly influences the determination of special points, such as the one affecting the argument of a complex number.