Problem 213 - A Complete Resource Book in Mathematics for JEE Main 2017

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 \(|z + \frac{1}{2}|\) is \(\frac{3}{2}\), not matching exactly with the options.

Step 1: Understand the Given Information

We are given a complex number \( z \) such that its absolute value is \(|z| \geq 2 \). Our objective is to find the minimum value of the expression \( \left|z + \frac{1}{2} \right| \).

Step 2: Use the Triangle Inequality

Recall the triangle inequality for complex numbers, which states that: \[ |z + w| \geq ||z| - |w||, \]where \( z \) and \( w \) are complex numbers. Let \( w = -\frac{1}{2} \). Hence, \[ |z + \frac{1}{2}| \geq ||z| - |-\frac{1}{2}|| = \left| |z| - \frac{1}{2} \right|. \]

Step 3: Substitute the Given Condition

Since \(|z| \geq 2\), the expression becomes:\[ |z + \frac{1}{2}| \geq \left| |z| - \frac{1}{2} \right|. \]The minimum value occurs when \[ |z| = 2. \]

Step 4: Calculate the Minimum Value

By substituting \(|z| = 2\) into the expression:\[ \left| 2 - \frac{1}{2} \right| = \frac{3}{2}. \] Thus, the minimum value of \(|z + \frac{1}{2}|\) is \( \frac{3}{2} \).

Step 5: Review the Options

Compare the calculated minimum value, which is \( \frac{3}{2} \), with the given options:- Option A: \( \frac{5}{2} \) is incorrect.- Option B: Interval (1, 2), incorrect because \( \frac{3}{2} \) is not completely in this interval.- Option C: Strictly greater than \( \frac{5}{2} \), incorrect.- Option D: Strictly greater than \( \frac{3}{2} \) is incorrect because the minimum is exactly \( \frac{3}{2} \).Hence, none of the options state the minimum correctly as exactly \( \frac{3}{2} \).

Key Concepts

Triangle Inequality

The triangle inequality is a crucial concept when dealing with complex numbers. Think of it like the sides of a triangle. In simple terms, the measure of one side of a triangle can never be greater than the sum of the other two sides. Translating this to complex numbers: if you have two complex numbers, say \( z \) and \( w \), the inequality says \( |z + w| \leq |z| + |w| \).
This principle can be reversed to illustrate that the difference in absolute values of two complex numbers is never less than the absolute value of their sum, expressed mathematically as \( |z + w| \geq ||z| - |w|| \).
This second form is precisely what we used in the original problem to assist in determining the minimum value of \( |z + \frac{1}{2}| \). It's important to grasp this fundamental tool, as it provides insight into the behavior of sums of complex numbers.

Minimum Value

Finding the minimum value of a complex expression often requires understanding the conditions given in the problem. Here, we had \(|z| \geq 2\), which is the starting point. Essentially, we focused on the lowest point \(|z|\) can reach under this inequality, which is when \(|z| = 2\).
Using the triangle inequality, we found that \(|z + \frac{1}{2}| \geq \left| |z| - \frac{1}{2} \right|\). Substituting the minimum value of \(|z|\) led to the expression \( \left| 2 - \frac{1}{2} \right| \), simplifying to \( \frac{3}{2} \).
This value represents the smallest possible value of the original expression when \( |z| \) is at its lowest permissible threshold, thus clarifying the correct result of finding the minimum value.

Absolute Value of Complex Numbers

Complex numbers have a real part and an imaginary part, written as \( z = a + bi \). The absolute value or modulus of a complex number is a measure of its size or magnitude. It's calculated using the formula \( |z| = \sqrt{a^2 + b^2} \).
The absolute value gives us a sense of distance from the origin in the complex plane, which is why it’s called the modulus. In practical terms, it's the length of the vector from the point representing the complex number back to the origin.
In the problem, the condition \( |z| \geq 2 \) tells us that the complex number is at least 2 units away from the origin, setting a fixed distance from which we can calculate additional relationships or transformations like \( z + \frac{1}{2} \). Understanding these distances helps solve various problems in complex analysis.