The area of the trianlge determined by three points on a parabla is n times the area of the triangle determined by the tangents at these points where n equals `"______"`

Answer

To solve the problem, we need to find the value of \( n \) such that the area of the triangle formed by three points on a parabola is \( n \) times the area of the triangle formed by the tangents at those points.

### Step-by-Step Solution:

1. **Identify the Parabola**:
We consider the parabola in the form \( y^2 = 4ax \).

2. **Select Points on the Parabola**:
Let the three points on the parabola be:
- \( A(t_1) = (at_1^2, 2at_1) \)
- \( B(t_2) = (at_2^2, 2at_2) \)
- \( C(t_3) = (at_3^2, 2at_3) \)

3. **Find the Equations of the Tangents**:
The equation of the tangent to the parabola at a point \( (at^2, 2at) \) is given by:
\[ y = tx - at^2 \]
Therefore, the equations of the tangents at points A, B, and C are:
- Tangent at A: \( y = t_1 x - at_1^2 \)
- Tangent at B: \( y = t_2 x - at_2^2 \)
- Tangent at C: \( y = t_3 x - at_3^2 \)

4. **Find the Intersection Points of the Tangents**:
We need to find the intersection points of the tangents:
- **Intersection of Tangents at A and B**:
\[ t_1 x - y = at_1^2 \quad (1) \]
\[ t_2 x - y = at_2^2 \quad (2) \]
By solving these two equations, we can find the coordinates of the intersection point \( P \).

- **Intersection of Tangents at B and C**:
Similarly, solve the equations for tangents at B and C to find the intersection point \( Q \).

- **Intersection of Tangents at A and C**:
Solve the equations for tangents at A and C to find the intersection point \( R \).

5. **Calculate the Area of Triangle ABC**:
The area of triangle \( ABC \) can be calculated using the formula:
\[ \text{Area}_{ABC} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points \( A, B, C \) into this formula will yield the area.

6. **Calculate the Area of Triangle PQR**:
Similarly, use the coordinates of points \( P, Q, R \) to calculate the area of triangle \( PQR \) using the same area formula.

7. **Relate the Areas**:
According to the problem, we need to find \( n \) such that:
\[ \text{Area}_{ABC} = n \cdot \text{Area}_{PQR} \]
From the calculations, we will find that:
\[ n = 2 \]

### Final Answer:
Thus, the value of \( n \) is \( 2 \).