If the length of focal chord of `y^2=4a x` is `l ,` then find the angle between the axis of the parabola and the focal chord.

Explanation

To solve the problem, we need to find the angle between the axis of the parabola \(y^2 = 4ax\) and a focal chord of length \(L\).

### Step-by-Step Solution:

1. **Understanding the Parabola**:
The given parabola is \(y^2 = 4ax\). The focus of this parabola is at the point \((a, 0)\).

**Hint**: Remember that the focus of the parabola \(y^2 = 4ax\) is located at \((a, 0)\).

2. **Focal Chord Properties**:
A focal chord is a line segment that passes through the focus and has its endpoints on the parabola. If a focal chord makes an angle \(\alpha\) with the axis of the parabola, we can use the formula for the length of the focal chord.

**Hint**: Recall that the length of a focal chord can be expressed in terms of the angle it makes with the axis.

3. **Length of the Focal Chord**:
The length \(L\) of the focal chord that makes an angle \(\alpha\) with the axis is given by:
\[ L = 4a \cos^2 \alpha \]

**Hint**: This formula relates the length of the focal chord to the angle it makes with the axis.

4. **Setting up the Equation**:
From the above formula, we can rearrange it to find \(\cos^2 \alpha\):
\[ \cos^2 \alpha = \frac{L}{4a} \]

**Hint**: Isolate \(\cos^2 \alpha\) to express it in terms of \(L\) and \(a\).

5. **Finding \(\sin^2 \alpha\)**:
Using the identity \(\sin^2 \alpha + \cos^2 \alpha = 1\), we can find \(\sin^2 \alpha\):
\[ \sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \frac{L}{4a} \]

**Hint**: Use the Pythagorean identity to find \(\sin^2 \alpha\).

6. **Calculating \(\sin \alpha\)**:
Taking the square root gives:
\[ \sin \alpha = \pm \sqrt{1 - \frac{L}{4a}} \]

**Hint**: Remember to consider both the positive and negative roots when taking the square root.

7. **Finding the Angle \(\alpha\)**:
Finally, we can express the angle \(\alpha\) in terms of \(L\) and \(a\):
\[ \alpha = \sin^{-1}\left(\sqrt{1 - \frac{L}{4a}}\right) \]

**Hint**: Use the inverse sine function to express the angle.

### Final Answer:
The angle \(\alpha\) between the axis of the parabola and the focal chord is given by:
\[ \alpha = \sin^{-1}\left(\sqrt{1 - \frac{L}{4a}}\right) \]