Question

Given that \( F=\dfrac{|z_1+z_2|}{|z_1|+|z_2|} \), where \(z_1=2+3i\) and \(z_2=-2+3i\) with \(i=\sqrt{-1}\), which one of the following options is CORRECT?

• A. \(F<0\)
• B. \(F<1\)
• C. \(F>1\)
• D. \(F=1\)

Hint:

- Use \(|a+bi|=\sqrt{a^2+b^2}\).
- Often, checking whether a ratio is \(<1\) or \(>1\) is faster than computing exact decimals.
- Symmetry like \(|z_1|=|z_2|\) can simplify sums in denominators.

Answer 1

The Correct Option is B

Step 1: Compute the sum: \[ z_1 + z_2 = (2-2) + (3+3)i = 6i \;\;\Rightarrow\;\; |z_1+z_2| = |6i| = 6. \]

Step 2: Compute the magnitudes: \[ |z_1| = \sqrt{2^2+3^2} = \sqrt{13}, \quad |z_2| = \sqrt{(-2)^2+3^2} = \sqrt{13}. \]

Step 3: Evaluate \(F\): \[ F = \frac{|z_1+z_2|}{|z_1|+|z_2|} = \frac{6}{\sqrt{13} + \sqrt{13}} = \frac{6}{2\sqrt{13}} = \frac{3}{\sqrt{13}} \approx 0.832 < 1. \] Hence \(0 < F < 1\).

Final Answer: \[ \boxed{F < 1} \]