Angle between tangents drawn at ends of focal chord of parabola `y^(2)` = 4ax is

Options

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

Answer

To find the angle between the tangents drawn at the ends of a focal chord of the parabola \(y^2 = 4ax\), we can follow these steps:

### Step 1: Identify the points on the parabola
For the parabola \(y^2 = 4ax\), the points at the ends of the focal chord can be represented in parametric form as:
- Point \(A(t_1) = (at_1^2, 2at_1)\)
- Point \(B(t_2) = (at_2^2, 2at_2)\)

### Step 2: Use the property of focal chords
A focal chord is a line segment that passes through the focus of the parabola. For the parabola \(y^2 = 4ax\), the focus is at the point \((a, 0)\). The relationship between the parameters \(t_1\) and \(t_2\) for points on a focal chord is given by:
\[ t_1 t_2 = -1 \]

### Step 3: Find the slopes of the tangents at points A and B
The slope of the tangent at any point \((x_1, y_1)\) on the parabola \(y^2 = 4ax\) can be calculated using the derivative:
\[ \frac{dy}{dx} = \frac{2a}{y} \]
For point \(A(t_1)\):
\[ \text{slope of tangent at } A = \frac{2a}{2at_1} = \frac{1}{t_1} \]
For point \(B(t_2)\):
\[ \text{slope of tangent at } B = \frac{2a}{2at_2} = \frac{1}{t_2} \]

### Step 4: Calculate the angle between the two tangents
The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Substituting \(m_1 = \frac{1}{t_1}\) and \(m_2 = \frac{1}{t_2}\):
\[ \tan \theta = \left| \frac{\frac{1}{t_1} - \frac{1}{t_2}}{1 + \frac{1}{t_1 t_2}} \right| \]
Since \(t_1 t_2 = -1\):
\[ \tan \theta = \left| \frac{\frac{1}{t_1} - \frac{1}{t_2}}{1 - 1} \right| \]
This simplifies to:
\[ \tan \theta = \left| \frac{t_2 - t_1}{t_1 t_2} \right| = \left| \frac{t_2 - t_1}{-1} \right| = |t_1 - t_2| \]

### Step 5: Determine the angle
Since \(t_1 t_2 = -1\), we can conclude that the angle between the tangents at the ends of the focal chord is:
\[ \theta = \frac{\pi}{2} \]

### Final Answer
The angle between the tangents drawn at the ends of the focal chord of the parabola \(y^2 = 4ax\) is \(\frac{\pi}{2}\).

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