If `A_1 B_2` and `A_2 B_2` are two focal chords of the parabola `y^2 = 4ax` then the chords `A_1 A_2` and `B_1 B_2` intersect on

Options

A. directrix
B. axis
C. T.V.
D. none

Answer

To solve the problem, we need to find the intersection point of the chords \( A_1 A_2 \) and \( B_1 B_2 \) for the parabola given by the equation \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Identify Points on the Parabola:**
- The points \( A_1 \) and \( A_2 \) are focal chords of the parabola. For a parabola \( y^2 = 4ax \), the coordinates of points on the parabola can be expressed in terms of the parameter \( t \).
- Let \( A_1 = (at_1^2, 2at_1) \) and \( A_2 = (at_2^2, 2at_2) \).
- The corresponding points \( B_1 \) and \( B_2 \) on the parabola can be calculated using the properties of focal chords:
- \( B_1 = \left( \frac{a}{t_1^2}, -\frac{2a}{t_1} \right) \)
- \( B_2 = \left( \frac{a}{t_2^2}, -\frac{2a}{t_2} \right) \)

2. **Equation of Line \( A_1 A_2 \):**
- The slope of the line joining points \( A_1 \) and \( A_2 \) is given by:
\[ \text{slope} = \frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2a(t_2 - t_1)}{a(t_2^2 - t_1^2)} = \frac{2(t_2 - t_1)}{t_2^2 - t_1^2} \]
- The equation of the line \( A_1 A_2 \) can be written as:
\[ y - 2at_1 = \frac{2(t_2 - t_1)}{t_2^2 - t_1^2}(x - at_1^2) \]

3. **Equation of Line \( B_1 B_2 \):**
- Similarly, the slope of the line joining points \( B_1 \) and \( B_2 \) is:
\[ \text{slope} = \frac{-\frac{2a}{t_2} + \frac{2a}{t_1}}{\frac{a}{t_2^2} - \frac{a}{t_1^2}} = \frac{-2a\left(\frac{1}{t_2} - \frac{1}{t_1}\right)}{a\left(\frac{1}{t_2^2} - \frac{1}{t_1^2}\right)} \]
- The equation of the line \( B_1 B_2 \) can be written as:
\[ y + \frac{2a}{t_1} = \frac{-2\left(\frac{1}{t_2} - \frac{1}{t_1}\right)}{\frac{1}{t_2^2} - \frac{1}{t_1^2}}\left(x - \frac{a}{t_1^2}\right) \]

4. **Finding Intersection Point:**
- To find the intersection point of the two lines, we set the equations equal to each other and solve for \( x \) and \( y \).
- After simplifying, we will find that the intersection point lies on the directrix of the parabola, which is given by the equation \( x + a = 0 \).

5. **Conclusion:**
- Therefore, the intersection of the chords \( A_1 A_2 \) and \( B_1 B_2 \) lies on the directrix of the parabola, which is the line \( x + a = 0 \).