Question

If $P_1P_2$ and $P_3P_4$ are two focal chords of the parabola $y^2 = 4ax$, then the chords $P_1P_3$ and $P_2P_4$ intersect on the

• A. directrix of the parabola
• B. axis of the parabola
• C. latus- rectum of the parabola
• D. y- axis

Answer 1

The Correct Option is A

We are given the parabola \( y^2 = 4ax \), and two focal chords \( P_1P_2 \) and \( P_3P_4 \). The points \( P_1, P_2, P_3, P_4 \) are points on the parabola, and the line joining two points on a parabola is called a chord. A focal chord is a chord that passes through the focus of the parabola.

Step 1: Properties of Focal Chords
For the parabola \( y^2 = 4ax \), the focus is at \( (a, 0) \), and the directrix is the line \( x = -a \). One important property of the parabola is that the product of the slopes of any two points on a focal chord is constant. In fact, if \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \) are points on a focal chord, the relationship between the coordinates of these points satisfies: \[ x_1 x_2 = a^2 \] This property holds for any pair of points on the focal chord. 

Step 2: Intersection of Chords
The chords \( P_1P_3 \) and \( P_2P_4 \) intersect at a point that has a special geometric property: they intersect on the directrix of the parabola. This result is a well-known property of the geometry of parabolas and follows from the focus-directrix definition of the parabola. 

Conclusion:
Therefore, the chords \( P_1P_3 \) and \( P_2P_4 \) intersect on the directrix of the parabola.

Answer:

\[ \boxed{\text{Directrix of the parabola}} \]