Combined equation of pair of tangent to the hyperbola `x^(2) - y^(2) = 8 " is " 8x^(2) - 7y^(2) - 16x + 8 = 0 ` and equation of chord of contect is `x = lambda` , then value of `lambda` is __________

Answer

To solve the problem, we need to find the value of \( \lambda \) given the combined equation of the pair of tangents to the hyperbola \( x^2 - y^2 = 8 \) and the equation of the chord of contact \( x = \lambda \).

### Step-by-Step Solution:

1. **Write the combined equation of the pair of tangents**:
The given equation is:
\[ 8x^2 - 7y^2 - 16x + 8 = 0 \]

2. **Rearrange the equation**:
Rearranging the equation gives:
\[ 8x^2 - 16x + 8 = 7y^2 \]

3. **Factor out common terms**:
Factor out 8 from the left side:
\[ 8(x^2 - 2x + 1) = 7y^2 \]

4. **Complete the square**:
The expression \( x^2 - 2x + 1 \) can be rewritten as:
\[ (x - 1)^2 \]
Thus, we have:
\[ 8(x - 1)^2 = 7y^2 \]

5. **Rearranging the equation**:
This can be rewritten as:
\[ (x - 1)^2 = \frac{7}{8}y^2 \]

6. **Finding the equations of the tangents**:
From the equation above, we can derive the equations of the tangents:
\[ x - 1 = \pm \sqrt{\frac{7}{8}}y \]
This gives us two equations:
\[ x + \sqrt{\frac{7}{8}}y = 1 \quad \text{(1)} \]
\[ x - \sqrt{\frac{7}{8}}y = 1 \quad \text{(2)} \]

7. **Finding the intersection point of the tangents**:
To find the intersection point of these two tangents, we can add equations (1) and (2):
\[ (x + \sqrt{\frac{7}{8}}y) + (x - \sqrt{\frac{7}{8}}y) = 1 + 1 \]
This simplifies to:
\[ 2x = 2 \implies x = 1 \]

8. **Substituting back to find y**:
Substitute \( x = 1 \) into either equation (1) or (2). Using equation (1):
\[ 1 + \sqrt{\frac{7}{8}}y = 1 \implies \sqrt{\frac{7}{8}}y = 0 \implies y = 0 \]
Thus, the intersection point is \( (1, 0) \).

9. **Using the point to find the chord of contact**:
The equation of the chord of contact is given by \( t = 0 \) where \( (x_1, y_1) = (1, 0) \) is the point of contact. The equation of the chord of contact for the hyperbola \( x^2 - y^2 = 8 \) is:
\[ \frac{x \cdot 1}{8} - \frac{y \cdot 0}{8} = 1 \]
Simplifying this gives:
\[ x = 8 \]

10. **Comparing with the given chord of contact**:
We know that the equation of the chord of contact is \( x = \lambda \). By comparing:
\[ \lambda = 8 \]

### Final Answer:
The value of \( \lambda \) is \( \boxed{8} \).