Question

The lengths of the two focal chords of the parabola \( y^2 = 16x \) is 25 units each. If these two chords cut the parabola at \( A, B, C, D \), then the area (in sq. units) of the quadrilateral formed by \( A, B, C, D \) is:

• A. \( \frac{625}{2} \)
• B. \( 180 \)
• C. \( 150 \)
• D. \( 300 \)

Hint:

In a parabola, the area formed by points on two focal chords can be computed using: \[ \text{Area} = 2a^2(t_1 - t_2)^2 \] where \( t_1, t_2 \) are the parameters of intersection points.

Answer 1

The Correct Option is D

For a parabola \( y^2 = 4ax \), the focal chord length formula in terms of parameters \( t_1, t_2 \) is: \[ L = a(t_1 - t_2)^2 \] For \( y^2 = 16x \), we have \( a = 4 \). Given that the chord lengths are 25, we solve: \[ 4(t_1 - t_2)^2 = 25 \] \[ (t_1 - t_2)^2 = \frac{25}{4} \] Thus, using the standard area formula for a quadrilateral formed by two focal chords: \[ \text{Area} = 2a^2(t_1 - t_2)^2 \] \[ = 2(4)^2 \times \frac{25}{4} \] \[ = 300 \text{ sq. units} \]