If `a x+b y=1` is tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` , then `a^2-b^2` is equal to (a)`1/(a^2e^2)` (b) `a^2e^2` (c)`b^2e^2` (d) none of these

Options

A. `1//a^(2)e^(2)`
B. `a^(2)e^(2)`
C. `b^(2)e^(2)` none of these
D. none of these

Explanation

To solve the problem, we need to find the value of \( a^2 - b^2 \) given that the line \( ax + by = 1 \) is tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

### Step-by-Step Solution:

1. **Identify the Tangent Line and Hyperbola:**
The equation of the tangent line is given as:
\[ ax + by = 1 \]
The equation of the hyperbola is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

2. **Standard Form of Tangent to Hyperbola:**
The standard equation of the tangent to the hyperbola in parametric form is:
\[ \frac{x \sec \theta}{a} - \frac{y \tan \theta}{b} = 1 \]

3. **Comparing Coefficients:**
We can compare the coefficients of \( x \) and \( y \) from both equations. From the tangent line \( ax + by = 1 \), we have:
- Coefficient of \( x \): \( \frac{\sec \theta}{a} = a \)
- Coefficient of \( y \): \( -\frac{\tan \theta}{b} = b \)

4. **Expressing Secant and Tangent:**
From the equations obtained, we can express:
\[ \sec \theta = a^2 \quad \text{and} \quad -\tan \theta = b^2 \]

5. **Using the Identity:**
We know that:
\[ 1 + \tan^2 \theta = \sec^2 \theta \]
Substituting the values we have:
\[ 1 + (-b^2)^2 = (a^2)^2 \]
This simplifies to:
\[ 1 + b^4 = a^4 \]

6. **Rearranging the Equation:**
Rearranging gives us:
\[ a^4 - b^4 = 1 \]

7. **Factoring the Difference of Squares:**
We can factor \( a^4 - b^4 \) as:
\[ (a^2 + b^2)(a^2 - b^2) = 1 \]

8. **Finding \( a^2 - b^2 \):**
We know that the eccentricity \( e \) of the hyperbola is given by:
\[ e^2 = 1 + \frac{b^2}{a^2} \]
Rearranging gives:
\[ e^2 = \frac{a^2 + b^2}{a^2} \]
Therefore, we can express \( a^2 + b^2 \) as:
\[ a^2 + b^2 = a^2 e^2 \]

9. **Substituting Back:**
Now substituting \( a^2 + b^2 \) back into our factored equation:
\[ (a^2 e^2)(a^2 - b^2) = 1 \]
This leads to:
\[ a^2 - b^2 = \frac{1}{a^2 e^2} \]

### Conclusion:
Thus, the value of \( a^2 - b^2 \) is:
\[ \boxed{\frac{1}{a^2 e^2}} \]