If `z_(1), z_(2), z_(3)` are 3 distinct complex such that `(3)/(|z_(1)-z_(2)|)=(5)/(|z_(2)-z_(3)|)=(7)/(|z_(3)-z_(1)|)`, then the value of `(9barz_(3))/(z_(1)-z_(2))+(25barz_(1))/(z_(2)-z_(3))+(49barz_(2))/(z_(3)-z_(1))` is equal to

Options

A. 0
B. 1
C. `-1`
D. 15

Answer

To solve the problem, we need to find the value of the expression given the conditions on the complex numbers \( z_1, z_2, z_3 \).

### Step-by-Step Solution:

1. **Set Up the Ratios**:
We start with the condition given in the problem:
\[ \frac{3}{|z_1 - z_2|} = \frac{5}{|z_2 - z_3|} = \frac{7}{|z_3 - z_1|} = k \]
From this, we can express the distances in terms of \( k \):
\[ |z_1 - z_2| = \frac{3}{k}, \quad |z_2 - z_3| = \frac{5}{k}, \quad |z_3 - z_1| = \frac{7}{k} \]

2. **Square the Distances**:
We will square each of the distances:
\[ |z_1 - z_2|^2 = \left(\frac{3}{k}\right)^2 = \frac{9}{k^2}, \quad |z_2 - z_3|^2 = \left(\frac{5}{k}\right)^2 = \frac{25}{k^2}, \quad |z_3 - z_1|^2 = \left(\frac{7}{k}\right)^2 = \frac{49}{k^2} \]

3. **Use the Property of Complex Numbers**:
We can relate the squared distances to the complex numbers:
\[ |z_1 - z_2|^2 = (z_1 - z_2)(\overline{z_1 - z_2}) = (z_1 - z_2)(\overline{z_1} - \overline{z_2}) \]
Thus, we can write:
\[ \frac{9}{k^2} = (z_1 - z_2)(\overline{z_1} - \overline{z_2}), \quad \frac{25}{k^2} = (z_2 - z_3)(\overline{z_2} - \overline{z_3}), \quad \frac{49}{k^2} = (z_3 - z_1)(\overline{z_3} - \overline{z_1}) \]

4. **Set Up the Expression to Evaluate**:
We need to evaluate:
\[ \frac{9 \overline{z_3}}{z_1 - z_2} + \frac{25 \overline{z_1}}{z_2 - z_3} + \frac{49 \overline{z_2}}{z_3 - z_1} \]

5. **Combine the Terms**:
We can rewrite the expression using the previously derived relationships:
\[ = \frac{9 \overline{z_3}}{z_1 - z_2} + \frac{25 \overline{z_1}}{z_2 - z_3} + \frac{49 \overline{z_2}}{z_3 - z_1} \]

6. **Factor Out Common Terms**:
We can factor out \( k^2 \) from the right-hand side of the equations we derived earlier:
\[ = k^2 \left( \frac{9}{|z_1 - z_2|} + \frac{25}{|z_2 - z_3|} + \frac{49}{|z_3 - z_1|} \right) \]

7. **Evaluate the Sum**:
Notice that the terms will cancel out because of the cyclic nature of the distances:
\[ \frac{9}{|z_1 - z_2|} + \frac{25}{|z_2 - z_3|} + \frac{49}{|z_3 - z_1|} = 0 \]
Thus, the entire expression evaluates to:
\[ = 0 \]

### Final Answer:
The value of the expression is:
\[ \boxed{0} \]