Let the tangents PQ and PR are drawn to `y^(2)=4ax` from any point P on the line `x+4a=0`. The angle subtended by the chord of contact QR at the vertex of the parabola `y^(2)=4ax` is

Options

A. `(pi)/(4)`
B. `(pi)/(3)`
C. `(pi)/(2)`
D. `(pi)/(6)`

Answer

To solve the problem, we need to find the angle subtended by the chord of contact QR at the vertex of the parabola \( y^2 = 4ax \). Let's break down the solution step by step:

### Step 1: Identify the Points on the Parabola
Let \( Q \) and \( R \) be points on the parabola \( y^2 = 4ax \). In parametric form, we can represent these points as:
- \( Q = (at_1^2, 2at_1) \)
- \( R = (at_2^2, 2at_2) \)

### Step 2: Determine the Point P on the Given Line
The point \( P \) lies on the line \( x + 4a = 0 \). We can express the coordinates of point \( P \) as:
- \( P = (-4a, y_P) \)

### Step 3: Find the Condition for Point P
Since \( P \) lies on the line, we can substitute \( x = -4a \) into the equation of the line:
- \( -4a + 4a = 0 \) which is satisfied for any \( y_P \).

### Step 4: Use the Chord of Contact
The chord of contact from point \( P \) to the parabola is given by the equation:
\[ yy_P = 2a(x + 4a) \]
This is the equation of the tangents drawn from point \( P \) to the parabola.

### Step 5: Find the Slopes of Lines OQ and OR
The vertex \( O \) of the parabola is at \( (0, 0) \). We can find the slopes of lines \( OQ \) and \( OR \):
- Slope of \( OQ \):
\[ \text{slope of } OQ = \frac{2at_1 - 0}{at_1^2 - 0} = \frac{2}{t_1} \]
- Slope of \( OR \):
\[ \text{slope of } OR = \frac{2at_2 - 0}{at_2^2 - 0} = \frac{2}{t_2} \]

### Step 6: Find the Angle Between the Two Lines
The angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) can be found using the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Substituting \( m_1 = \frac{2}{t_1} \) and \( m_2 = \frac{2}{t_2} \):
\[ \tan \theta = \left| \frac{\frac{2}{t_1} - \frac{2}{t_2}}{1 + \frac{2}{t_1} \cdot \frac{2}{t_2}} \right| \]

### Step 7: Substitute the Product of Slopes
We know from the problem that \( t_1 t_2 = -4 \). Thus:
\[ m_1 m_2 = \frac{2}{t_1} \cdot \frac{2}{t_2} = \frac{4}{t_1 t_2} = \frac{4}{-4} = -1 \]
This means that the lines \( OQ \) and \( OR \) are perpendicular.

### Conclusion
Since the product of the slopes \( m_1 \) and \( m_2 \) is \(-1\), the angle subtended by the chord of contact QR at the vertex O is \( 90^\circ \).