Let PQ be a focal chord of the parabola `y^(2)=4ax`. The tangents to the parabola at P and Q meet at point lying on the line
`y=2x+a,alt0`.
If chord PQ subtends an angle `theta` at the vertex of `y^(2)=4ax`, then `tantheta=`

Options

A. `2sqrt(7)//3`
B. `-2sqrt(7)//3`
C. `2sqrt(5)//3`
D. `-2sqrt(5)//3`

Explanation

To solve the problem, we need to find the value of \( \tan \theta \) where \( \theta \) is the angle subtended by the focal chord \( PQ \) at the vertex of the parabola \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Identify Points on the Parabola:**
Let \( P \) and \( Q \) be points on the parabola \( y^2 = 4ax \). We can represent these points using parameters \( t \) and \( -\frac{1}{t} \):
- For point \( P \): \( P(at^2, 2at) \)
- For point \( Q \): \( Q\left(a\left(-\frac{1}{t}\right)^2, 2a\left(-\frac{1}{t}\right)\right) = \left(\frac{a}{t^2}, -\frac{2a}{t}\right) \)

2. **Find the Coordinates of Points \( P \) and \( Q \):**
- \( P = (at^2, 2at) \)
- \( Q = \left(\frac{a}{t^2}, -\frac{2a}{t}\right) \)

3. **Find the Slopes of Tangents at Points \( P \) and \( Q \):**
The slope of the tangent at point \( P \) is given by the derivative of the parabola:
\[ \text{slope at } P = \frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t} \]
The slope of the tangent at point \( Q \) is:
\[ \text{slope at } Q = \frac{dy}{dx} = \frac{-2a/t}{2a/(t^2)} = -t \]

4. **Equation of Tangents at Points \( P \) and \( Q \):**
The equation of the tangent at \( P \) is:
\[ y - 2at = \frac{1}{t}(x - at^2) \implies y = \frac{1}{t}x + 2at - \frac{at^2}{t} \]
The equation simplifies to:
\[ y = \frac{1}{t}x + 2a - at \]

The equation of the tangent at \( Q \) is:
\[ y + \frac{2a}{t} = -t\left(x - \frac{a}{t^2}\right) \implies y = -tx + \frac{a}{t} + \frac{2a}{t} \]
This simplifies to:
\[ y = -tx + \frac{3a}{t} \]

5. **Find the Intersection of the Tangents:**
Set the two equations equal to find the intersection point \( R \):
\[ \frac{1}{t}x + 2a - at = -tx + \frac{3a}{t} \]
Rearranging gives:
\[ \left(t + \frac{1}{t}\right)x = \frac{3a}{t} - 2a + at \]
Solve for \( x \) and then substitute back to find \( y \).

6. **Find the Angle \( \theta \):**
The angle \( \theta \) subtended by the chord \( PQ \) at the vertex can be found using the slopes of the tangents:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Where \( m_1 = \frac{1}{t} \) and \( m_2 = -t \).

Substitute the values:
\[ \tan \theta = \left| \frac{\frac{1}{t} + t}{1 - 1} \right| \text{ (undefined, indicating a right angle)} \]

### Conclusion:
Since the tangents at points \( P \) and \( Q \) meet at a right angle, we conclude that \( \tan \theta = 1 \).