Problem 136 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(|z-4+3 i| \leq 2\), then the least and the greatest values of \(|z|\) are (A) 3,7 (B) 4,7 (C) 3,9 (D) None of these

Step-by-Step Solution

Answer

The least and greatest values of \(|z|\) are 3 and 7, respectively (Option A).

Step 1: Interpret the given inequality

The inequality \(|z - 4 + 3i| \leq 2\) represents all points inside or on the boundary of a circle in the complex plane. The center of the circle is the complex number \(4 - 3i\), and the radius is \(2\).

Step 2: Express z in the complex plane

Let \(z = x + yi\) be a complex number where \(x, y\) are real numbers. We need to find the least and greatest values of \(|z| = \sqrt{x^2 + y^2}\).

Step 3: Analyze geometric position

The distance from the center \(4 - 3i\) to the origin \(0\) is found using the formula for distance between two points: \(d = \sqrt{(4-0)^2 + (-3-0)^2} = \sqrt{16 + 9} = 5\).

Step 4: Describe the circle's influence on distance

The circle extends 2 units in all directions around its center \(4 - 3i\). Thus, the minimum distance from any point on this circle to the origin is \(5 - 2 = 3\) and the maximum distance is \(5 + 2 = 7\).

Step 5: Deduce the least and greatest values of |z|

Since the circle centered at \(4 - 3i\) of radius 2 meets the origin at these maximum and minimum distances, \(|z|\) will range from 3 to 7. Therefore, the least value is 3 and the greatest value is 7.

Key Concepts

Inequalities

Understanding inequalities in mathematics is essential when dealing with complex numbers. An inequality like \(|z - 4 + 3i| \leq 2\) represents a set of points in the complex plane. These points satisfy the condition that their distance to the point (or center) at \(4 - 3i\) is less than or equal to 2. In other words, these points lie inside or on the boundary of a circle.Key Points:

• The inequality describes a spatial region, not just a line or a single point.
• All complex numbers \(z\) satisfying the inequality will have a magnitude difference governed by the condition, which here is 2.
• This forms a circle with a particular radius, helping visualize the set of solutions in the complex plane.

Geometric Interpretation

In complex numbers, geometric interpretation plays a vital role as it helps one visualize and solve inequalities and equations. Imagine the complex plane as a flat, two-dimensional field where every complex number corresponds to a specific point. For instance, a complex number \(4 - 3i\) represents a point where 4 is on the real (horizontal) axis and -3 is on the imaginary (vertical) axis.When we interpret \(|z - 4 + 3i| \leq 2\), it depicts all the points lying within or on a circle. Here, \(z\) is moving around, and its permissible movement is restricted by this circular boundary centered at \(4 - 3i\). This visual representation makes it easier to assess the relationship between these points and understand possible solutions in terms of magnitudes.

• Circle Center: The fixed point given, like \(4 - 3i\), becomes the center.
• Radius: The value given alongside the inequality determines how far from the center points can be, here it's 2.
• Real and Imaginary Axes: Help determine the position and orientation of the points on this imaginary plane.

Distance Formula

The distance formula is a mathematical tool used to calculate how far apart two points are in a plane. In the complex plane, which maps real and imaginary numbers, this formula helps find the distance between the origin \(0\) and any complex number like \(4 - 3i\).To calculate this, the distance \((d)\) from a point \(a + bi\) to the origin uses the formula \( d = \sqrt{(a - 0)^2 + (b - 0)^2}). \)For the given point \(4 - 3i\), it works out as follows:- Start by calculating the differences along each axis, which are 4 (real part) and -3 (imaginary part).- Square these values and sum them: \(16 + 9 = 25\).- Find the square root of the sum to get the distance: \(\sqrt{25} = 5\).Therefore, the point \(4 - 3i\) is 5 units away from the origin, which becomes crucial when accounting for additional distances resulting from the circle's radius.

Circle in Complex Plane

A circle in the complex plane is a geometrical configuration where all points within a specific distance (radius) from a central point are included. This set of points is defined by the equation \(|z - a| \leq r\), where \(a\) is the center and \(r\) is the radius.For our exercise, it is important to:

• Identify the Circle's Parameters: Here, the center is \(4 - 3i\) and the radius is 2. These give the circle its position and size.
• Determine Critical Distances: Calculate from the center to the extrema, such as closest and farthest points from the origin. Max distance occurs when traveling the radius outward from the center, adding 2 to the distance of 5 from center to origin, totaling 7. The minimum distance occurs inward, reducing the same distance by 2, totaling 3.
• Solution Application: These distances directly answer the problem, showing that the least and greatest values of \(|z|\) accordingly are 3 and 7.

Through understanding these concepts, identifying these points becomes operational, completing the puzzle of finding ranges of \(|z|\) calculated for the specified inequality.