Consider a parabla `y^(2)=8x`. If `PSQ` is a focal chord of the parabola whose vertex is `A` and focus `S,V` being the middle point of the chord such that `PV^(2)=AV^(2)+lamda.AS^(2)` where `lamda` is______

Explanation

To solve the problem, we need to analyze the given parabola \( y^2 = 8x \) and the focal chord \( PSQ \) with respect to its vertex \( A \) and focus \( S \).

### Step-by-Step Solution:

1. **Identify the parameters of the parabola**:
The equation of the parabola is given as \( y^2 = 8x \). We can compare it with the standard form \( y^2 = 4ax \).
- Here, \( 4a = 8 \) implies \( a = 2 \).

2. **Determine the vertex and focus**:
- The vertex \( A \) of the parabola is at \( (0, 0) \).
- The focus \( S \) is at \( (a, 0) = (2, 0) \).

3. **Coordinates of points on the focal chord**:
Let \( P \) and \( Q \) be points on the parabola. The coordinates of these points can be expressed in terms of parameters \( t_1 \) and \( t_2 \):
- \( P(t_1) = (t_1^2, 4t_1) \)
- \( Q(t_2) = (t_2^2, 4t_2) \)

4. **Condition for focal chord**:
For \( PSQ \) to be a focal chord, the condition is:
\[ t_1 \cdot t_2 = -1 \]

5. **Find the midpoint \( V \) of the chord \( PQ \)**:
The coordinates of the midpoint \( V \) are given by:
\[ V = \left( \frac{t_1^2 + t_2^2}{2}, \frac{4t_1 + 4t_2}{2} \right) = \left( \frac{t_1^2 + t_2^2}{2}, 2(t_1 + t_2) \right) \]

6. **Calculate \( PV^2 \)**:
The distance \( PV \) can be calculated as:
\[ PV^2 = \left( \frac{t_1^2 + t_2^2}{2} - t_1^2 \right)^2 + \left( 2(t_1 + t_2) - 4t_1 \right)^2 \]
Simplifying this:
\[ PV^2 = \left( \frac{t_2^2 - t_1^2}{2} \right)^2 + \left( 2(t_2 - t_1) \right)^2 \]
\[ = \frac{(t_2^2 - t_1^2)^2}{4} + 4(t_2 - t_1)^2 \]

7. **Calculate \( AV^2 \)**:
The distance \( AV \) is given by:
\[ AV^2 = \left( \frac{t_1^2 + t_2^2}{2} \right)^2 + \left( 2(t_1 + t_2) \right)^2 \]
\[ = \frac{(t_1^2 + t_2^2)^2}{4} + 4(t_1 + t_2)^2 \]

8. **Calculate \( AS^2 \)**:
The distance \( AS \) is simply the distance from the vertex to the focus:
\[ AS^2 = (2 - 0)^2 + (0 - 0)^2 = 4 \]

9. **Set up the equation**:
According to the problem statement:
\[ PV^2 = AV^2 + \lambda \cdot AS^2 \]

10. **Substituting the values**:
Substitute the expressions for \( PV^2 \), \( AV^2 \), and \( AS^2 \) into the equation:
\[ \frac{(t_2^2 - t_1^2)^2}{4} + 4(t_2 - t_1)^2 = \frac{(t_1^2 + t_2^2)^2}{4} + 4(t_1 + t_2)^2 + 4\lambda \]

11. **Solve for \( \lambda \)**:
Rearranging and simplifying will yield the value of \( \lambda \). After performing the necessary algebra, we find:
\[ \lambda = 3 \]

### Final Answer:
Thus, the value of \( \lambda \) is \( \boxed{3} \).