The tangent at a point `P` on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` meets one of the directrix at `Fdot` If `P F` subtends an angle `theta` at the corresponding focus, then `theta=` `pi/4` (b) `pi/2` (c) `(3pi)/4` (d) `pi`

Options

A. `pi//4`
B. `pi//2`
C. `3pi//4`
D. `pi`

Explanation

To solve the problem step by step, we will analyze the hyperbola and the properties of the tangent line at a point on the hyperbola.

### Step 1: Understand the Hyperbola
The equation of the hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
This hyperbola opens along the x-axis with foci at \( (ae, 0) \) and \( (-ae, 0) \), where \( e = \sqrt{1 + \frac{b^2}{a^2}} \).

**Hint:** Recall the definition of the foci and directrices of a hyperbola.

### Step 2: Determine the Point \( P \)
Let \( P \) be a point on the hyperbola, which can be represented in parametric form as:
\[ P(a \sec \theta, b \tan \theta) \]
where \( \theta \) is the parameter.

**Hint:** Use the parametric equations of the hyperbola to find the coordinates of point \( P \).

### Step 3: Equation of the Tangent Line at Point \( P \)
The equation of the tangent line at point \( P(a \sec \theta, b \tan \theta) \) can be derived using the formula for the tangent to a hyperbola:
\[ \frac{x \sec \theta}{a} - \frac{y \tan \theta}{b} = 1 \]

**Hint:** Remember the formula for the tangent line to a hyperbola at a given point.

### Step 4: Find the Directrix
The directrix of the hyperbola is given by the equation:
\[ x = \frac{a}{e} \]
where \( e \) is the eccentricity of the hyperbola.

**Hint:** Recall the relationship between the directrix and the foci of the hyperbola.

### Step 5: Intersection of the Tangent and Directrix
To find the point \( F \) where the tangent meets the directrix, substitute \( x = \frac{a}{e} \) into the tangent equation:
\[ \frac{\frac{a}{e} \sec \theta}{a} - \frac{y \tan \theta}{b} = 1 \]
This simplifies to:
\[ \frac{\sec \theta}{e} - \frac{y \tan \theta}{b} = 1 \]
From this, we can solve for \( y \) to find the coordinates of point \( F \).

**Hint:** Substitute the value of \( x \) from the directrix into the tangent equation and solve for \( y \).

### Step 6: Angle Subtended at the Focus
The angle \( \theta \) subtended by line segment \( PF \) at the focus can be found using the slopes of the lines \( PF \) and the line from the focus to point \( P \). The slopes can be calculated from the coordinates of \( P \) and \( F \).

**Hint:** Use the formula for the angle between two lines given their slopes.

### Step 7: Determine the Value of \( \theta \)
After calculating the angle, we find that:
\[ \theta = \frac{\pi}{2} \]
Thus, the correct option is (b) \( \frac{\pi}{2} \).

**Hint:** Verify your calculations to ensure that the angle corresponds to one of the given options.

### Final Answer
The angle \( \theta \) subtended at the focus by the line segment \( PF \) is:
\[ \theta = \frac{\pi}{2} \]