Consider the following two statements : Statement I : For any two non-zero complex numbers z1, z2, (

Q: Wait, in Statement I, are we allowed to just multiply by $(|z_1| + |z_2|)$? Does that change the inequality sign?

Great question! Since $|z_1|$ and $|z_2|$ are magnitudes of non-zero complex numbers, their sum is strictly positive. Multiplying an inequality by a positive number preserves the direction of the inequality. It's perfectly safe!

Q: Why does Statement II result in 0? I thought it might be 1 because of the symmetry.

Symmetry is a powerful tool, but it doesn't guarantee a specific value like 1. When you expand the terms using conjugates, you get a telescoping sum of conjugates: $(\bar{y} - \bar{z}) + (\bar{z} - \bar{x}) + (\bar{x} - \bar{y})$. Everything cancels out perfectly to zero. It's a beautiful algebraic collapse!

Q: Do I need to know the coordinates of $x, y, z$ to solve Statement II?

Not at all! That's the elegance of complex number algebra. By treating them as abstract points and using the conjugate property, the specific coordinates become irrelevant. The result is independent of the triangle's shape.

Question

Consider the following two statements : 
Statement I : For any two non-zero complex numbers $z_1, z_2$, $(|z_1| + |z_2|) \left| \frac{z_1}{|z_1|} + \frac{z_2}{|z_2|} ight| \le 2 (|z_1| + |z_2|)$, and 
Statement II : If $x, y, z$ are three distinct complex numbers and $a, b, c$ are three positive real numbers such that $\frac{a}{|y-z|} = \frac{b}{|z-x|} = \frac{c}{|x-y|}$, then $\frac{a^2}{y-z} + \frac{b^2}{z-x} + \frac{c^2}{x-y} = 1$. 
Between the above two statements,

Options:

(A) Statement I is correct but Statement II is incorrect.

(B) both Statement I and Statement II are correct.

(C) both Statement I and Statement II are incorrect.

(D) Statement I is incorrect but Statement II is correct.

Sigma Percentile · Accepted Answer

### Analyzing the Unit Vector Insight

Let us look at Statement I: $(|z_1| + |z_2|) \left| \frac{z_1}{|z_1|} + \frac{z_2}{|z_2|} \right| \le 2 (|z_1| + |z_2|)$.

Define $u_1 = \frac{z_1}{|z_1|}$ and $u_2 = \frac{z_2}{|z_2|}$. These are unit vectors residing on the unit circle, meaning their magnitudes are fixed at exactly $1$.

The inequality simplifies to $|u_1 + u_2| \le 2$. This is a direct application of the **Triangle Inequality**, which states that the magnitude of the sum of two vectors is less than or equal to the sum of their individual magnitudes.

Since $|u_1| = 1$ and $|u_2| = 1$, we have:

$$|u_1 + u_2| \le |u_1| + |u_2| = 1 + 1 = 2$$

Multiplying both sides by the positive real number $(|z_1| + |z_2|)$ preserves the inequality. Thus, **Statement I is correct**.

### The Geometric Trap

Now, consider Statement II. We are given three distinct complex numbers $x, y, z$ and positive real numbers $a, b, c$ such that:

$$\frac{a}{|y-z|} = \frac{b}{|z-x|} = \frac{c}{|x-y|} = \sqrt{\lambda}$$

This implies the following relationships for the squares of the constants:

$$a^2 = \lambda |y-z|^2, \quad b^2 = \lambda |z-x|^2, \quad c^2 = \lambda |x-y|^2$$

We must evaluate the expression $\frac{a^2}{y-z} + \frac{b^2}{z-x} + \frac{c^2}{x-y}$.

### The Algebraic Resolution

Recall the fundamental property of complex numbers: the square of the magnitude is the product of the number and its conjugate, $|w|^2 = w \bar{w}$.

Applying this to the first term, we get:

$$a^2 = \lambda (y-z)(\bar{y}-\bar{z})$$

Dividing by $(y-z)$ yields $\lambda (\bar{y} - \bar{z})$. By symmetry, the other terms in the expression simplify to $\lambda (\bar{z} - \bar{x})$ and $\lambda (\bar{x} - \bar{y})$.

Summing these terms results in:

$$\lambda [(\bar{y} - \bar{z}) + (\bar{z} - \bar{x}) + (\bar{x} - \bar{y})] = \lambda [0] = 0$$

The sum collapses to zero due to the telescoping nature of the conjugates. Since Statement II claims the sum is $1$, **Statement II is incorrect**.

Select Answer:

Visualized Solution

Analyzing Statement I

Unit Vectors in Complex Plane

Applying Triangle Inequality

Simplifying the Inequality

Concluding Statement I

Analyzing Statement II

Setting up a Constant

Using Conjugate Properties

Rearranging the First Term

Expressions for Other Terms

Summing the Terms

Final Verdict

The Sigma Insight: Algebraic Operations on Complex Numbers

Analyzing the Unit Vector Insight

The Geometric Trap

The Algebraic Resolution

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