The length of a focal chord of the parabola `y^2=4ax` making an angle `theta` with the axis of the parabola is `(a> 0)`is`:`

Options

A. `4a"cosec"^(2)theta`
B. `4a cos theta "cosec"^(2)theta`
C. `4a cot theta "cosec"^(2)theta`
D. `2a" cosec"^(2)theta`

Answer

To find the length of a focal chord of the parabola \( y^2 = 4ax \) making an angle \( \theta \) with the axis of the parabola, we can follow these steps:

### Step 1: Understand the parabola and focal chord
The equation of the parabola is given as \( y^2 = 4ax \). The focus of this parabola is at the point \( (a, 0) \). A focal chord is a line segment that passes through the focus and has its endpoints on the parabola.

### Step 2: Identify the points on the parabola
Let \( P(t) \) be a point on the parabola represented in parametric form:
- The coordinates of point \( P \) are \( (at^2, 2at) \).
- The coordinates of the point \( Q \) on the focal chord can be represented as \( (a/t^2, -2a/t) \).

### Step 3: Calculate the slope of the focal chord
The slope of the line segment \( PQ \) can be calculated using the coordinates of points \( P \) and \( Q \):
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2a/t - 2at}{a/t^2 - at^2} \]
This simplifies to:
\[ \text{slope} = \frac{2a(-t^2 - 1/t)}{a(1/t^2 - t^2)} = \frac{2(-t^2 - 1/t)}{1/t^2 - t^2} \]

### Step 4: Relate the slope to the angle \( \theta \)
The slope of the line can also be expressed in terms of the angle \( \theta \):
\[ \tan(\theta) = \text{slope} \]
Thus, we can set up the equation:
\[ \tan(\theta) = \frac{2(-t^2 - 1/t)}{1/t^2 - t^2} \]

### Step 5: Simplify the equation
From the previous step, we can rearrange and simplify to find a relationship involving \( t \) and \( \theta \). After simplifications, we can express:
\[ t - \frac{1}{t} = 2 \cot(\theta) \]

### Step 6: Find \( t + \frac{1}{t} \)
Using the identity \( (a + b)^2 = (a - b)^2 + 4ab \):
\[ \left(t + \frac{1}{t}\right)^2 = \left(t - \frac{1}{t}\right)^2 + 4 \]
Substituting \( t - \frac{1}{t} = 2 \cot(\theta) \):
\[ \left(t + \frac{1}{t}\right)^2 = (2 \cot(\theta))^2 + 4 = 4 \cot^2(\theta) + 4 = 4(\cot^2(\theta) + 1) = 4 \csc^2(\theta) \]

### Step 7: Calculate the length of the focal chord
The length of the focal chord can be expressed as:
\[ \text{Length} = a \left(t + \frac{1}{t}\right) = a \cdot \sqrt{4 \csc^2(\theta)} = 2a \cdot 2 \csc(\theta) = 4a \csc(\theta) \]

### Conclusion
Thus, the length of the focal chord of the parabola \( y^2 = 4ax \) making an angle \( \theta \) with the axis of the parabola is:
\[ \boxed{4a \csc(\theta)} \]