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

Options

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

Answer

To solve the problem, we start with the given condition involving the complex numbers \( z_1, z_2, z_3 \):

\[ \frac{3}{|z_1 - z_2|} = \frac{5}{|z_2 - z_3|} = \frac{7}{|z_3 - z_1|} = k \]

### Step 1: Express the distances in terms of \( k \)

From the above equality, we can express the distances as follows:

1. \( |z_1 - z_2| = \frac{3}{k} \)
2. \( |z_2 - z_3| = \frac{5}{k} \)
3. \( |z_3 - z_1| = \frac{7}{k} \)

### Step 2: Square the equations

Next, we square each of these equations:

1. \( |z_1 - z_2|^2 = \left(\frac{3}{k}\right)^2 = \frac{9}{k^2} \)
2. \( |z_2 - z_3|^2 = \left(\frac{5}{k}\right)^2 = \frac{25}{k^2} \)
3. \( |z_3 - z_1|^2 = \left(\frac{7}{k}\right)^2 = \frac{49}{k^2} \)

### Step 3: Use the properties of complex numbers

Using the property of complex numbers, we can express the squared distances in terms of the complex numbers:

1. \( |z_1 - z_2|^2 = (z_1 - z_2)(\overline{z_1 - z_2}) = (z_1 - z_2)(\overline{z_1} - \overline{z_2}) \)
2. \( |z_2 - z_3|^2 = (z_2 - z_3)(\overline{z_2 - z_3}) = (z_2 - z_3)(\overline{z_2} - \overline{z_3}) \)
3. \( |z_3 - z_1|^2 = (z_3 - z_1)(\overline{z_3 - z_1}) = (z_3 - z_1)(\overline{z_3} - \overline{z_1}) \)

### Step 4: Substitute back into the equations

We can rewrite the equations as:

1. \( 9 = k^2 |z_1 - z_2|^2 \)
2. \( 25 = k^2 |z_2 - z_3|^2 \)
3. \( 49 = k^2 |z_3 - z_1|^2 \)

### Step 5: Find the expression we need

We need to evaluate the expression:

\[ \frac{9}{z_1 - z_2} + \frac{25}{z_2 - z_3} + \frac{49}{z_3 - z_1} \]

### Step 6: Substitute the values

Using the earlier results, we can substitute:

\[ \frac{9}{z_1 - z_2} = k^2 \cdot \frac{z_1 - z_2}{9} \]
\[ \frac{25}{z_2 - z_3} = k^2 \cdot \frac{z_2 - z_3}{25} \]
\[ \frac{49}{z_3 - z_1} = k^2 \cdot \frac{z_3 - z_1}{49} \]

### Step 7: Combine the expressions

Combining these gives:

\[ k^2 \left( \frac{z_1 - z_2}{9} + \frac{z_2 - z_3}{25} + \frac{z_3 - z_1}{49} \right) \]

### Step 8: Simplify the expression

Notice that:

\[ \frac{z_1 - z_2}{9} + \frac{z_2 - z_3}{25} + \frac{z_3 - z_1}{49} = 0 \]

This is because the terms cancel out due to the cyclic nature of the expression.

### Conclusion

Thus, the entire expression evaluates to:

\[ k^2 \cdot 0 = 0 \]

So, the final answer is:

\[ \boxed{0} \]