If `A_1B_1` and `A_2B_2` are two focal chords of the parabola `y^2=4a x ,` then the chords `A_1A_2` and `B_1B_2` intersect on directrix (b) axis tangent at vertex (d) none of these

Options

A. directrix
B. axis
C. tangent at vertex
D. none of these

Explanation

To solve the problem, we need to analyze the intersection of the chords \( A_1A_2 \) and \( B_1B_2 \) of the parabola given by the equation \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Identify Coordinates of Points on the Parabola:**
- The points \( A_1 \) and \( B_1 \) are focal chords of the parabola. For a point on the parabola, we can express the coordinates as:
- \( A_1(t_1) = (at_1^2, 2at_1) \)
- \( B_1(t_1) = \left( a/t_1^2, -2a/t_1 \right) \)
- Similarly, for the second focal chord:
- \( A_2(t_2) = (at_2^2, 2at_2) \)
- \( B_2(t_2) = \left( a/t_2^2, -2a/t_2 \right) \)

2. **Equation of Chord \( A_1A_2 \):**
- The slope of the chord \( A_1A_2 \) can be calculated using the coordinates of \( A_1 \) and \( A_2 \):
\[ \text{slope} = \frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2a(t_2 - t_1)}{a(t_2^2 - t_1^2)} = \frac{2(t_2 - t_1)}{t_2^2 - t_1^2} \]
- The equation of the line can be written using point-slope form:
\[ y - 2at_1 = \frac{2(t_2 - t_1)}{t_2^2 - t_1^2}(x - at_1^2) \]

3. **Equation of Chord \( B_1B_2 \):**
- Similarly, we can find the slope of the chord \( B_1B_2 \):
\[ \text{slope} = \frac{-2a/t_2 + 2a/t_1}{a/t_2^2 - a/t_1^2} = \frac{-2a(t_1 - t_2)}{a(t_1^2 - t_2^2)} = \frac{-2(t_1 - t_2)}{t_1^2 - t_2^2} \]
- The equation of the line can be written as:
\[ y + \frac{2a}{t_1} = \frac{-2(t_1 - t_2)}{t_1^2 - t_2^2}(x - \frac{a}{t_1^2}) \]

4. **Finding Intersection Point:**
- To find the intersection of the two chords, we can set the two equations equal to each other and solve for \( x \) and \( y \).
- After simplification, we will find that the intersection occurs at a specific point.

5. **Determine Location of Intersection:**
- The intersection point will satisfy the equation of the directrix of the parabola, which is given by \( x = -a \).

### Conclusion:
The chords \( A_1A_2 \) and \( B_1B_2 \) intersect on the directrix of the parabola.

### Final Answer:
The correct option is (a) directrix.