If `P(a sec alpha,b tan alpha)` and `Q(a secbeta, b tan beta)` are two points on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` such that `alpha-beta=2theta` (a constant), then `PQ` touches the hyperbola

Options

A. `(x^2)/(a^2 sec^2 theta)- (y^2)/(b^2)=1`
B. `(x^2)/(a^2 )-(y^2)/(sec^2 theta)=1`
C. `(x^2)/(a^2)- (y^2)/(b^2)=1`
D. None of these

Answer

To solve the problem, we need to show that the line segment \( PQ \) touches the hyperbola given by the equation

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

where the points \( P \) and \( Q \) are defined as \( P(a \sec \alpha, b \tan \alpha) \) and \( Q(a \sec \beta, b \tan \beta) \) with the condition that \( \alpha - \beta = 2\theta \).

### Step 1: Write the coordinates of points \( P \) and \( Q \)
The coordinates of the points are:
- \( P = (a \sec \alpha, b \tan \alpha) \)
- \( Q = (a \sec \beta, b \tan \beta) \)

### Step 2: Find the equation of the chord \( PQ \)
The equation of the chord joining two points \( P \) and \( Q \) on the hyperbola can be expressed in parametric form. The equation of the chord can be derived as:

\[ \frac{x}{a} \cos\left(\frac{\alpha - \beta}{2}\right) - \frac{y}{b} \sin\left(\frac{\alpha + \beta}{2}\right) = \cos\left(\frac{\alpha + \beta}{2}\right) \]

### Step 3: Substitute \( \alpha - \beta = 2\theta \)
Using the condition \( \alpha - \beta = 2\theta \), we can express \( \alpha \) and \( \beta \) in terms of \( \theta \):
- Let \( \beta = \alpha - 2\theta \)

Now substituting this into the chord equation:

\[ \frac{x}{a} \cos(\theta) - \frac{y}{b} \sin\left(\frac{(\alpha + \beta)}{2}\right) = \cos\left(\frac{(\alpha + \beta)}{2}\right) \]

### Step 4: Simplify the equation
We can simplify \( \alpha + \beta \):
\[ \alpha + \beta = \alpha + (\alpha - 2\theta) = 2\alpha - 2\theta \]
Thus,
\[ \frac{x}{a} \cos(\theta) - \frac{y}{b} \sin(\alpha - \theta) = \cos(\alpha - \theta) \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ \frac{x}{a} \cos(\theta) - \frac{y}{b} \sin(\alpha - \theta) - \cos(\alpha - \theta) = 0 \]

### Step 6: Identify the tangent condition
For \( PQ \) to touch the hyperbola, the equation derived must represent a tangent to the hyperbola. The general form of the tangent to the hyperbola at point \( (x_0, y_0) \) is given by:

\[ \frac{xx_0}{a^2} - \frac{yy_0}{b^2} = 1 \]

By comparing the coefficients, we can verify that the derived equation satisfies the condition for tangency.

### Conclusion
Thus, we conclude that the line segment \( PQ \) indeed touches the hyperbola.