Let `lamdax – 2y =mu` be a tangent to the hyperbola `a^2x^2 - y^2=b^2`. Then `(lamda/a)^2 -(mu/b)^2` is equal to :

Options

A. `-2`
B. `-4`
C. 2
D. 4

Answer

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

### Step-by-Step Solution:

1. **Rewrite the Tangent Equation**:
The tangent line can be expressed as:
\[ \lambda x - 2y = \mu \]
Rearranging gives:
\[ 2y = \lambda x - \mu \quad \Rightarrow \quad y = \frac{\lambda}{2} x - \frac{\mu}{2} \]
Here, the slope \(m\) of the tangent is:
\[ m = \frac{\lambda}{2} \]

2. **Use the Slope Form of the Hyperbola**:
The slope form of the tangent to the hyperbola \(a^2 x^2 - y^2 = b^2\) is given by:
\[ y = mx \pm \sqrt{b^2(1 + m^2)} \]
Substituting \(m = \frac{\lambda}{2}\):
\[ y = \frac{\lambda}{2} x \pm \sqrt{b^2\left(1 + \left(\frac{\lambda}{2}\right)^2\right)} \]

3. **Equate the Two Expressions for y**:
Since both expressions represent the same line, we can equate them:
\[ \frac{\lambda}{2} x - \frac{\mu}{2} = \frac{\lambda}{2} x \pm \sqrt{b^2\left(1 + \left(\frac{\lambda}{2}\right)^2\right)} \]
This implies:
\[ -\frac{\mu}{2} = \pm \sqrt{b^2\left(1 + \left(\frac{\lambda}{2}\right)^2\right)} \]

4. **Square Both Sides**:
Squaring both sides gives:
\[ \left(-\frac{\mu}{2}\right)^2 = b^2\left(1 + \left(\frac{\lambda}{2}\right)^2\right) \]
Simplifying:
\[ \frac{\mu^2}{4} = b^2 + b^2\left(\frac{\lambda^2}{4}\right) \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{\mu^2}{4} - b^2 = \frac{b^2\lambda^2}{4} \]
Multiplying through by 4:
\[ \mu^2 - 4b^2 = b^2\lambda^2 \]

6. **Divide by \(b^2\)**:
Dividing the entire equation by \(b^2\) gives:
\[ \frac{\mu^2}{b^2} - 4 = \frac{\lambda^2}{b^2} \]

7. **Rearranging to Find the Desired Expression**:
Rearranging gives:
\[ \frac{\lambda^2}{b^2} = \frac{\mu^2}{b^2} - 4 \]
Therefore:
\[ \left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2 = 4 \]

### Final Answer:
\[ \left(\frac{\lambda}{a}\right)^2 - \left(\frac{\mu}{b}\right)^2 = 4 \]