Question

Consider the following two statements:
Statement I: For any two non-zero complex numbers \( z_1, z_2 \),
\((|z_1| + |z_2|) \left| \frac{z_1}{|z_1|} + \frac{z_2}{|z_2|} \right| \leq 2 (|z_1| + |z_2|)\)
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,

• A. both Statement I and Statement II are incorrect.
• B. Statement I is incorrect but Statement II is correct.
• C. Statement I is correct but Statement II is incorrect.
• D. both Statement I and Statement II are correct.

Answer 1

The Correct Option is C

Statement I:

\[ \left( \lvert z_1 \rvert + \lvert z_2 \rvert \right) \left\lvert \frac{z_1}{\lvert z_1 \rvert} + \frac{z_2}{\lvert z_2 \rvert} \right\rvert \leq 2 \left( \lvert z_1 \rvert + \lvert z_2 \rvert \right) \]

Since

\[ \left\lvert \frac{z_1}{\lvert z_1 \rvert} + \frac{z_2}{\lvert z_2 \rvert} \right\rvert \leq 2 \]

we have

\[ \left( \lvert z_1 \rvert + \lvert z_2 \rvert \right) \left\lvert \frac{z_1}{\lvert z_1 \rvert} + \frac{z_2}{\lvert z_2 \rvert} \right\rvert \leq 2 \left( \lvert z_1 \rvert + \lvert z_2 \rvert \right) \]

Thus, Statement I is correct.

Statement II: Given

\[ \frac{a}{\lvert y - z \rvert} = \frac{b}{\lvert z - x \rvert} = \frac{c}{\lvert x - y \rvert} \]

let

\[ \frac{a}{\lvert y - z \rvert} = \frac{b}{\lvert z - x \rvert} = \frac{c}{\lvert x - y \rvert} = \lambda \]

Then,

\[ a^2 = \lambda \lvert y - z \rvert, \quad b^2 = \lambda \lvert z - x \rvert, \quad c^2 = \lambda \lvert x - y \rvert \]

Substituting, we get:

\[ \frac{a^2}{y - z} + \frac{b^2}{z - x} + \frac{c^2}{x - y} = \lambda \left( \frac{y - z}{y - z} + \frac{z - x}{z - x} + \frac{x - y}{x - y} \right) \]
Thus, Statement II is false.