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 \( \lambda \) for the given hyperbola and the combined equation of the pair of tangents. Here's the step-by-step solution:

### Step 1: Identify the Hyperbola
The given hyperbola is:
\[ x^2 - y^2 = 8 \]
This can be rewritten in standard form as:
\[ \frac{x^2}{8} - \frac{y^2}{8} = 1 \]
This indicates that it is a rectangular hyperbola.

### Step 2: Write the Combined Equation of the Tangents
The combined equation of the pair of tangents to the hyperbola is given as:
\[ 8x^2 - 7y^2 - 16x + 8 = 0 \]
We can rearrange this equation to express it in terms of \( y^2 \):
\[ 8x^2 - 16x + 8 = 7y^2 \]
Dividing the entire equation by 8 gives:
\[ x^2 - 2x + 1 = \frac{7}{8}y^2 \]
This simplifies to:
\[ (x - 1)^2 = \frac{7}{8}y^2 \]

### Step 3: Find the Tangent Lines
Taking the square root of both sides, we get the equations of the tangents:
\[ x - 1 = \pm \sqrt{\frac{7}{8}}y \]
This leads to two tangent equations:
1. \( x - 1 = \sqrt{\frac{7}{8}}y \)
2. \( x - 1 = -\sqrt{\frac{7}{8}}y \)

### Step 4: Find the Point of Intersection
To find the point of intersection of these two tangents, we can add the two equations:
\[ (x - 1) + (x - 1) = \sqrt{\frac{7}{8}}y - \sqrt{\frac{7}{8}}y \]
This simplifies to:
\[ 2(x - 1) = 0 \implies x = 1 \]
Substituting \( x = 1 \) into either tangent equation to find \( y \):
\[ 1 - 1 = \sqrt{\frac{7}{8}}y \implies 0 = \sqrt{\frac{7}{8}}y \implies y = 0 \]
Thus, the point of intersection is \( (1, 0) \).

### Step 5: Write the Equation of the Chord of Contact
The equation of the chord of contact for the hyperbola \( x^2 - y^2 = 8 \) at the point \( (1, 0) \) is given by:
\[ t = 0 \quad \text{(where \( t \) is the parameter)} \]
Using the formula for the chord of contact:
\[ \frac{x \cdot 1}{8} - \frac{y \cdot 0}{8} = 1 \]
This simplifies to:
\[ x = 8 \]
Since we need to find \( \lambda \) such that \( x = \lambda \), we have:
\[ \lambda = 8 \]

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