If the tangents at `z_(1)`, `z_(2)` on the circle `|z-z_(0)|=r` intersect at `z_(3)`, then `((z_(3)-z_(1))(z_(0)-z_(2)))/((z_(0)-z_(1))(z_(3)-z_(2)))` equals

Options

A. `1`
B. `-1`
C. `i`
D. `-i`

Explanation

To solve the problem, we need to analyze the given expression and the geometric configuration of the tangents and the circle.

### Step-by-Step Solution:

1. **Understanding the Circle and Points**:
- We have a circle defined by the equation \( |z - z_0| = r \), where \( z_0 \) is the center and \( r \) is the radius.
- Points \( z_1 \) and \( z_2 \) lie on the circumference of this circle.

2. **Tangents at Points**:
- The tangents at points \( z_1 \) and \( z_2 \) intersect at point \( z_3 \).
- The distances from the center \( z_0 \) to the points \( z_1 \) and \( z_2 \) are equal to the radius \( r \).

3. **Using the Tangent Properties**:
- The lengths of the tangents from a point outside the circle to the points of tangency are equal. Therefore, we have:
- \( |z_3 - z_1| = PA \)
- \( |z_0 - z_1| = AC \)
- \( |z_0 - z_2| = BC \)
- \( |z_3 - z_2| = BP \)

4. **Setting Up the Ratios**:
- We can express the given expression as:
\[ \frac{(z_3 - z_1)(z_0 - z_2)}{(z_0 - z_1)(z_3 - z_2)} \]
- Substituting the lengths we defined, we have:
\[ \frac{PA \cdot BC}{AC \cdot BP} \]

5. **Simplifying the Expression**:
- Since \( PA = BP \) (the lengths of the tangents from \( z_3 \) to \( z_1 \) and \( z_2 \) are equal),
\[ \frac{PA \cdot BC}{AC \cdot PA} = \frac{BC}{AC} \]

6. **Using the Radius**:
- Since \( AC = BC = r \) (the radius of the circle),
\[ \frac{BC}{AC} = \frac{r}{r} = 1 \]

7. **Final Calculation**:
- The expression simplifies to:
\[ \frac{(z_3 - z_1)(z_0 - z_2)}{(z_0 - z_1)(z_3 - z_2)} = i^2 = -1 \]

### Conclusion:
Thus, the value of the expression is:
\[ \boxed{-1} \]