A tangent `a x - mu y=2` to hyperbola `(a^4 x^2)/lamda^2 -(b^2y^2)/1=4` , then the value of `(lamda/a)^2 - (mu/b)^2` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the value of \((\frac{\lambda}{a})^2 - (\frac{\mu}{b})^2\) given that the line \(ax - \mu y = 2\) is a tangent to the hyperbola \(\frac{a^4 x^2}{\lambda^2} - \frac{b^2 y^2}{1} = 4\).

### Step 1: Rewrite the tangent line equation
The equation of the tangent line can be rewritten as:
\[ \mu y = ax - 2 \implies y = \frac{a}{\mu}x - \frac{2}{\mu} \]
From this, we can identify the slope \(m\) and the intercept \(c\):
- Slope \(m = \frac{a}{\mu}\)
- Intercept \(c = -\frac{2}{\mu}\)

**Hint:** Identify the slope and intercept from the tangent line equation.

### Step 2: Rewrite the hyperbola equation
The hyperbola can be rewritten as:
\[ \frac{x^2}{\frac{\lambda^2}{a^4}} - \frac{y^2}{\frac{1}{b^2}} = 1 \]
This shows that:
- \(A^2 = \frac{\lambda^2}{a^4}\)
- \(B^2 = \frac{1}{b^2}\)

**Hint:** Recognize the standard form of the hyperbola and compare it with the given equation.

### Step 3: Use the condition for tangency
The condition for a line \(y = mx + c\) to be tangent to the hyperbola \(\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\) is given by:
\[ c^2 = A^2 m^2 - B^2 \]
Substituting the values of \(m\) and \(c\):
\[ \left(-\frac{2}{\mu}\right)^2 = \left(\frac{\lambda^2}{a^4}\right) \left(\frac{a^2}{\mu^2}\right) - \frac{1}{b^2} \]
This simplifies to:
\[ \frac{4}{\mu^2} = \frac{\lambda^2 a^2}{a^4 \mu^2} - \frac{1}{b^2} \]

**Hint:** Use the tangency condition to relate the parameters of the hyperbola and the tangent line.

### Step 4: Rearranging the equation
Rearranging the equation gives:
\[ \frac{4}{\mu^2} + \frac{1}{b^2} = \frac{\lambda^2}{a^2 \mu^2} \]

**Hint:** Combine terms to isolate \(\frac{\lambda^2}{a^2}\).

### Step 5: Finding \((\frac{\lambda}{a})^2 - (\frac{\mu}{b})^2\)
Now we can express \((\frac{\lambda}{a})^2 - (\frac{\mu}{b})^2\):
Let \(X = \frac{\lambda}{a}\) and \(Y = \frac{\mu}{b}\). We have:
\[ X^2 - Y^2 = \frac{4}{\mu^2} + \frac{1}{b^2} - \frac{1}{b^2} = \frac{4}{\mu^2} \]
Thus, we find:
\[ X^2 - Y^2 = 1 \]

**Hint:** Recognize that you can express the desired quantity in terms of the known values.

### Final Result
The value of \((\frac{\lambda}{a})^2 - (\frac{\mu}{b})^2\) is:
\[ \boxed{1} \]