Problem 137 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1}, z_{2}\) are two complex numbers and \(c>0\) such that \(\left|z_{1}+z_{2}\right|^{2} \leq(1+c)\left|z_{1}\right|^{2}+k\left|z_{2}\right|^{2}\), then \(k=\) (A) \(1-c\) (B) \(c-1\) (C) \(1+c^{-1}\) (D) \(1-c^{-1}\)

Step-by-Step Solution

Answer

The value of \(k\) is option (C) \(1+c^{-1}\).

Step 1: Expand the Left Hand Side

Start by expanding the left side of the inequality. We have \( |z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) = z_1 \overline{z_1} + z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_2}. \)This can be rewritten as \( |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}) \).

Step 2: Rewrite the Inequality

The original inequality is \( |z_1 + z_2|^2 \leq (1+c)|z_1|^2 + k|z_2|^2 \). Substitute the expanded form from Step 1: \( |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}) \leq (1+c)|z_1|^2 + k|z_2|^2 \).

Step 3: Rearrange the Terms

Rearrange the terms to compare the terms on both sides appropriately:\[ |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}) \leq |z_1|^2 + c|z_1|^2 + k|z_2|^2 \].Subtract \( |z_1|^2 \) and \( |z_2|^2 \):\[ 2 \text{Re}(z_1 \overline{z_2}) \leq c|z_1|^2 + (k-1)|z_2|^2 \].

Step 4: Use the Cauchy-Schwarz Inequality

Apply the Cauchy-Schwarz inequality to the term \(2 \text{Re}(z_1 \overline{z_2})\), giving \(2 \text{Re}(z_1 \overline{z_2}) \leq 2|z_1||z_2| \). Thus, the inequality becomes \( 2|z_1||z_2| \leq c|z_1|^2 + (k-1)|z_2|^2 \).

Step 5: Compare Coefficients

To satisfy the inequality for all complex numbers \(z_1\) and \(z_2\), balance the terms. To find the smallest possible value of \(k\), compare the terms:i) Assume equality where \( |z_1| = |z_2| \). Then, \(2|z_1||z_2| = 2|z_1|^2\).ii) Calculate \(c|z_1|^2 + (k-1)|z_2|^2 = c|z_1|^2 + (k-1)|z_1|^2\).Equating these gives: \(2|z_1|^2 = c|z_1|^2 + (k-1)|z_1|^2\), simplifying to:\(2 = c + (k-1)\).Solve for \(k\):\(k = 1 + 1/c\).

Step 6: Conclusion

By following the process outlined, we determined that \(k = 1 + 1/c\), which corresponds to option (C).

Key Concepts

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality is a fundamental concept in mathematics, particularly in linear algebra and analysis. It is incredibly useful when dealing with vectors and complex numbers. The inequality states that for any vectors \(\mathbf{u}\) and \(\mathbf{v}\) in an inner product space, the following holds: \[|\langle \mathbf{u}, \mathbf{v} \rangle|^2 \leq \langle \mathbf{u}, \mathbf{u} \rangle \cdot \langle \mathbf{v}, \mathbf{v} \rangle\]This inequality essentially provides a bound on the absolute value of the inner product of two vectors by the product of the magnitudes (or norms) of the vectors themselves.

• This inequality helps ensure that certain expressions stay within specific limits.
• It's instrumental in proofs in complex analysis and functional analysis.

In the context of complex numbers, the Cauchy-Schwarz inequality can help relate the magnitudes of two complex numbers and is used to simplify expressions that might otherwise be quite difficult to manage.

expanding expressions

Expanding expressions involves taking a compact mathematical statement and rewriting it in a more extended form. This can help clarify relationships between terms and is especially useful when dealing with polynomial expressions or complex numbers.For complex numbers, it's simple yet powerful. Consider the expression \(|z_1 + z_2|^2\). It can be expanded as follows: \[(z_1 + z_2)(\overline{z_1 + z_2}) = z_1 \overline{z_1} + z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_2}\]This further simplifies to:\[|z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2})\]

• It allows us to clearly see the interaction between terms, especially with the real part \(2 \text{Re}(z_1 \overline{z_2})\).
• Expanding expressions can help simplify complex inequalities and compare different sides of equations effectively.

By using such expansions, we can break down problems into simpler components that are easier to analyze and solve.

inequality manipulation

Inequality manipulation involves rearranging and altering inequalities to make them easier to solve or compare with other expressions. It is a widely used technique in mathematics, especially in calculus and algebra. It helps to detect underlying relationships between mathematical expressions.In the exercise, we start with:\[|z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}) \leq (1+c)|z_1|^2 + k|z_2|^2\]This inequality is rearranged by collecting similar terms and subtracting \(|z_1|^2\) and \(|z_2|^2\) from both sides to isolate terms involving the real part:\[2 \text{Re}(z_1 \overline{z_2}) \leq c|z_1|^2 + (k-1)|z_2|^2\]

• Rearranging terms can help us identify the importance of coefficients and constants such as \(c\) and \(k\).
• It allows for strategic comparisons that make solving for unknowns more straightforward.

Through careful manipulation, we achieved a form that naturally led us to apply the Cauchy-Schwarz inequality to determine bounds and ultimately solve for \(k\). By such manipulation, we can find solutions more easily and ensure our inequalities hold true for all possible values of the variables involved.