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 \( y^2 = 4ax \). Here are the steps to arrive at the solution:

### Step 1: Identify the point P
The point P lies on the line \( x + 4a = 0 \). This means that the x-coordinate of point P is \( -4a \). We can denote point P as \( P(-4a, y) \) where y can be any value.

### Step 2: Determine the tangents from point P to the parabola
The equation of the parabola is given by \( y^2 = 4ax \). The tangents from a point \( (x_1, y_1) \) to the parabola can be expressed using the formula:
\[ yy_1 = 2a(x + x_1) \]
Substituting \( x_1 = -4a \) into the equation gives:
\[ yy_1 = 2a(x - 4a) \]

### Step 3: Find the coordinates of points Q and R
Let the points of tangency be \( Q(t_1) \) and \( R(t_2) \) on the parabola. The coordinates of these points are:
\[ Q(t_1) = (at_1^2, 2at_1) \quad \text{and} \quad R(t_2) = (at_2^2, 2at_2) \]

### Step 4: Use the property of tangents
From the point P, the tangents intersect the parabola at points Q and R. The product of the slopes of the tangents can be determined using the relationship between the parameters \( t_1 \) and \( t_2 \):
\[ t_1 t_2 = -4 \]

### Step 5: Calculate the slopes of lines OQ and OR
The slope of line OQ (from the origin O to point Q) is:
\[ m_1 = \frac{2at_1}{at_1^2} = \frac{2}{t_1} \]
The slope of line OR (from the origin O to point R) is:
\[ m_2 = \frac{2at_2}{at_2^2} = \frac{2}{t_2} \]

### Step 6: Find the product of the slopes
The product of the slopes \( m_1 \) and \( m_2 \) is:
\[ m_1 \cdot m_2 = \left(\frac{2}{t_1}\right) \cdot \left(\frac{2}{t_2}\right) = \frac{4}{t_1 t_2} \]
Substituting \( t_1 t_2 = -4 \):
\[ m_1 \cdot m_2 = \frac{4}{-4} = -1 \]

### Step 7: Determine the angle between the lines
The product of the slopes being -1 indicates that the lines OQ and OR are perpendicular to each other. Therefore, the angle \( \theta \) subtended by the chord QR at the vertex O (0,0) is:
\[ \theta = \frac{\pi}{2} \]

### Conclusion
The angle subtended by the chord of contact QR at the vertex of the parabola \( y^2 = 4ax \) is \( \frac{\pi}{2} \).