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

Answer: (C)

Statement I is correct but Statement II is incorrect.

Answer 2

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.