Question

Let ABCD be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides AD and BC of the trapezium be parallel to the y-axis. If the diagonal AC is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of ABCD is:

• A. \( \frac{125}{8} \)
• B. \( \frac{75}{8} \)
• C. \( \frac{25}{2} \)
• D. \( \frac{75}{4} \)

Hint:

For geometric problems involving parabolas, always consider the symmetry and use the properties of the parabola.

Answer 1

The Correct Option is B

We are given the geometry of the trapezium and need to calculate its area. 

Step 1: First, determine the coordinates of the vertices of the trapezium using the equation \( y^2 = 4x \). 

Step 2: Calculate the length of diagonal AC by using the distance formula between the points. 

Step 3: Use the area formula for a trapezium, which involves calculating the parallel sides' lengths and height, to find the area. 

Final Conclusion: The area of ABCD is \( \frac{75}{8} \), which is Option 2.