Problem 137 - 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

(A) 3,7

Step 1: Understanding the Inequality

The inequality \( |z - 4 + 3i| \leq 2 \) is describing a region in the complex plane. This is a disk centered at \( 4 - 3i \) with a radius of 2.

Step 2: Visualize the Region

Visualize or sketch this region on the complex plane. The center of the circle is at \( (4, -3) \) with a radius stretching 2 units in all directions.

Step 3: Identifying |z|

For any complex number \( z = x + yi \), \( |z| = \sqrt{x^2 + y^2} \) represents its distance from the origin \( (0, 0) \) on the complex plane. We need to identify the farthest and closest points of the disk boundary from the origin.

Step 4: Find Closest Point to Origin

The closest point on the boundary of the disk \( |z - (4 - 3i)| = 2 \) to the origin can be found by calculating the distance from the origin to the center of the circle, then subtracting the radius: \ \\[\text{Distance from origin to center} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\] \ \Thus, the closest distance from the origin is \( 5 - 2 = 3 \).

Step 5: Find Farthest Point from Origin

The farthest point on the boundary of the disk from the origin can be found by adding the radius to the distance from the origin to the center: \ \\[\text{Farthest point from origin} = 5 + 2 = 7\] \ \This represents the greatest distance from the origin to any point on the boundary of the disk.

Key Concepts

Inequality in Complex Plane

When you see an inequality like \(|z - 4 + 3i| \leq 2\), it is referring to a region in the complex plane. Here, we're exploring a circular area centered around the point \(4 - 3i\) on the complex plane. The part \(|z - 4 + 3i|\) indicates the distance from any complex number \(z\) to the fixed point \((4, -3)\), and the inequality \(\leq 2\) describes a circle with a radius of 2. This means any complex number \(z\) that satisfies this inequality falls inside or on the circle surrounding \(4 - 3i\). This concept helps define a region containing all possible values \(z\) that lie within the given distance from the point \(4 - 3i\). Such visualizations are crucial in understanding complex inequalities as they provide a geometric interpretation of the solution set.

Geometric Representation of Complex Numbers

Complex numbers can be represented as points or vectors in the complex plane, which resembles a Cartesian coordinate system. Every complex number \(z = x + yi\) is associated with the point \((x, y)\). When an inequality suggests a circle, such as \(|z - 4 + 3i| \leq 2\), it means we are considering a circular area. The point \((4, -3)\) serves as the circle's center, extending outward with a radius of 2.

• The x-coordinate corresponds to the real part of the complex number.
• The y-coordinate corresponds to the imaginary part.

This delicious imagery helps grasp how complex numbers move, rotate, and cover space within a plane. It's important to internalize this picture as it simplifies many complex number operations that seem abstract in algebraic forms. By visualizing, we can easily handle tasks like identifying the boundary and internal points of specific regions within the complex plane.

Distance from Origin in Complex Plane

The concept of distance from the origin is significant when dealing with complex numbers. Every complex number \(z = x + yi\) has a magnitude or modulus, \(|z| = \sqrt{x^2 + y^2}\), measuring its distance from the origin \((0, 0)\). This acts similarly to how we calculate the distance of points in a two-dimensional space.

• To find the closest point on the circle to the origin, we compute the direct distance from the origin to the center of the circle and subtract the radius of 2.
• Conversely, for the farthest point, we add the radius to the distance from the center to origin.

In our circle, the center is at \(4 - 3i\). Thus, the original distance is \(\sqrt{4^2 + (-3)^2} = 5\). Therefore, the nearest point from the origin on the circle's edge is \(5 - 2 = 3\), and the furthest is \(5 + 2 = 7\). Understanding this helps tackle problems related to distance estimation, circle boundaries, and understanding the limits of complex regions.