Problem 213 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z\) is a complex number such that \(|z| \geq 2\), then the minimum value of \(\left|z+\frac{1}{2}\right|\) [2014] (A) is equal to \(\frac{5}{2}\) (B) lies in the interval \((1,2)\) (C) is strictly greater than \(\frac{5}{2}\) (D) is strictly greater than \(\frac{3}{2}\) but less than \(\frac{5}{2}\)

Step-by-Step Solution

Answer

The minimum value of \(\left|z+\frac{1}{2}\right|\) is strictly greater than \(\frac{3}{2}\) but less than \(\frac{5}{2}\); it's (D).

Step 1: Understand the Complex Number Condition

The given condition is \(|z| \geq 2\), which means the complex number \(z\) is located outside or on the boundary of a circle with radius 2 and center at the origin \((0,0)\) in the complex plane.

Step 2: Express the Given Expression

We need to evaluate the minimum value of \(\left| z + \frac{1}{2} \right|\). This represents the distance from the point \(-\frac{1}{2}\) on the real axis to the point \(z\) on the complex plane.

Step 3: Geometrical Interpretation of Distances

To find the minimum distance, consider \(z\) on the boundary of the circle \(|z| = 2\). The point \(z\) will be closest to \(-\frac{1}{2}\) when the line segment linking the center of the circle \((0,0)\) and \(z\) also passes through \(-\frac{1}{2}\).

Step 4: Calculate the Closest Point

By geometry, the closest point on the circle \(|z| = 2\) to \(-\frac{1}{2}\) will lie on the line joining \(-\frac{1}{2}\) and the origin. This point can be calculated as \(-\frac{1}{2} - 0\) = \(-\frac{1}{2}\) in direction extended to a point \(z\) such that the entire distance from the origin is 2. Using the line equation, parameterize \(z = re^{i\theta}\) to determine it physically on the circle such that the radial distance remains \(2\).

Step 5: Calculate the Minimum Value

Using the geometric distance method directly between \(-\frac{1}{2}\) and boundary of circle, express the distance formula: \(d = \sqrt{\left(2 + \frac{1}{2} \right)^2 + 0^2} = \frac{3}{2}\).

Step 6: Conclusion

The minimum value of \(\left| z + \frac{1}{2} \right|\) is therefore strictly greater than \(\frac{3}{2}\) but less than \(\frac{5}{2}\); hence the correct choice is (D).

Key Concepts

Minimum Distance

The concept of minimum distance comes into play when we are asked to find the smallest possible gap between two points or sets on the complex plane. In this exercise, we need to determine the minimum distance from the point \(-\frac{1}{2}\) on the real axis to any point \(z\) located on or outside a given circle centered at the origin with a radius of 2. To solve this, imagine the point \(-\frac{1}{2}\) on the complex plane. We measure the shortest path to a point \(z\) that satisfies \(|z| \geq 2\). This point occurs where the line segment originating from this specific point is closest to the circle's circumference in a straight path. As such, the minimum span results in a specific geometric configuration, leading us to conclude that the minimum distance is more than \(-\frac{3}{2}\) and less than \(\frac{5}{2}\).

Geometric Interpretation

Visualizing complex numbers and their properties in geometric terms helps clarify their relationships. This involves translating algebraic conditions, such as the modulus condition \(|z| = r\), into geometric regions on the plane. Here, \(z\) represents complex numbers on or beyond a circle in the complex plane with radius 2. When looking at the distance to another point, \(-\frac{1}{2}\), we interpret this as a quest to find the closest location on the circle to intersect or align with the path to the origin, hopping onto the shortest distance line connecting these two. This geometric configuration effectively simplifies our arithmetic understanding of the problem by providing a visual,at once intuitive explanation of it.

Complex Plane

The complex plane is a two-dimensional plane for representing complex numbers. It comprises a horizontal axis (representing real parts) and a vertical axis (representing imaginary parts). In our problem, \(z = x + yi\) is placed within this plane, showcasing its position and distance relative to other points, a vital aid in solving geometric queries like minimum distance. Here, visualize the circle \(|z| = 2\), perfectly encircling the origin. The dots \(z\)'s, which can be any complex number satisfying the modulus condition, dwell anywhere in this circle's shaded area or on its edge. Manipulating the complex plane allows easy graphical interpretation of complex algebraic problems, rendering them more digestible.

Circle on Complex Plane

A circle in the complex plane is defined by all the complex numbers with a fixed modulus relative to its center point, here centered at the origin. For our given problem, the circle defined by \(|z| = 2\) contains all numbers equidistant from its center, offering a circle with radius 2. Graphically conceptualizing this circle, any computation or operation stays within or along its boundary, depending on the given conditions. This set-up showcases the central role of understanding and utilizing the circle's geometry to derive distance solutions to problems involving it and other points in the surrounding space on the complex plane. Here, finding the minimum distance involves positioning or calculating other points relative to this symmetric and perfectly round figure that cloaks the origin.