If `(2z_(1))/(3z_(2))` is purely imaginary number, then `|(z_(1)-z_(2))/(z_(1)+z_(2))|^(4)` is equal to

Options

A. `3//2`
B. 1
C. `2//3`
D. `4//9`

Answer

To solve the problem, we need to analyze the given condition and calculate the required expression step by step.

### Step 1: Understand the Given Condition
We are given that \(\frac{2z_1}{3z_2}\) is purely imaginary. This means that the real part of this expression is zero.

### Step 2: Express the Condition Mathematically
From the condition, we can express it as:
\[ \frac{2z_1}{3z_2} = ki \quad \text{(where \(k\) is a real number)} \]
This implies:
\[ 2z_1 = 3z_2 \cdot ki \]
From this, we can express \(z_1\) in terms of \(z_2\):
\[ z_1 = \frac{3k}{2} z_2 i \]

### Step 3: Calculate the Required Expression
We need to find:
\[ \left| \frac{z_1 - z_2}{z_1 + z_2} \right|^4 \]

### Step 4: Substitute \(z_1\) in the Expression
Substituting \(z_1\):
\[ \frac{z_1 - z_2}{z_1 + z_2} = \frac{\frac{3k}{2} z_2 i - z_2}{\frac{3k}{2} z_2 i + z_2} \]
Factoring out \(z_2\) from both the numerator and the denominator:
\[ = \frac{z_2 \left(\frac{3k}{2} i - 1\right)}{z_2 \left(\frac{3k}{2} i + 1\right)} = \frac{\frac{3k}{2} i - 1}{\frac{3k}{2} i + 1} \]

### Step 5: Calculate the Modulus
Now we need to find the modulus:
\[ \left| \frac{\frac{3k}{2} i - 1}{\frac{3k}{2} i + 1} \right| \]
Using the property of modulus:
\[ = \frac{\left| \frac{3k}{2} i - 1 \right|}{\left| \frac{3k}{2} i + 1 \right|} \]

### Step 6: Calculate the Moduli
Calculate the modulus of the numerator:
\[ \left| \frac{3k}{2} i - 1 \right| = \sqrt{\left(0\right)^2 + \left(\frac{3k}{2}\right)^2 + 1^2} = \sqrt{\frac{9k^2}{4} + 1} \]
And for the denominator:
\[ \left| \frac{3k}{2} i + 1 \right| = \sqrt{\left(0\right)^2 + \left(\frac{3k}{2}\right)^2 + 1^2} = \sqrt{\frac{9k^2}{4} + 1} \]

### Step 7: Simplify the Expression
Thus, we have:
\[ \left| \frac{z_1 - z_2}{z_1 + z_2} \right| = \frac{\sqrt{\frac{9k^2}{4} + 1}}{\sqrt{\frac{9k^2}{4} + 1}} = 1 \]

### Step 8: Raise to the Power of 4
Finally, we raise this to the power of 4:
\[ \left| \frac{z_1 - z_2}{z_1 + z_2} \right|^4 = 1^4 = 1 \]

### Conclusion
Thus, the value of \(\left| \frac{z_1 - z_2}{z_1 + z_2} \right|^4\) is equal to \(1\).