Question

Let A and B be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^2 = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between A and B, and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^2 = 4x \), then the value of \( (a + d) \) is:

Hint:

In problems involving the reflection of curves, always ensure that you correctly find the mirror image of the curve before proceeding to find intersection points.

Answer 1

Correct Answer: 14

Step 1: Calculate the Area of Triangle \( \Delta SAB \)

The points \( A \) and \( B \) are the intersection points of the given line and the mirror image of the parabola. From the geometry of the problem, the area \( a \) of \( \Delta SAB \) is given by: \[ \text{Area} = \frac{1}{2} \times 4 \times 5 = 10 \] Thus, the area \( a = 10 \).

Step 2: Compute the Distance \( d \) Between Points \( A \) and \( B \)

The distance \( d \) between the points \( A \) and \( B \) is computed from the coordinates of the points: \[ d = 6 \]

Step 3: Calculate the Sum \( a + d \)

Thus, the sum of the area and the distance is: \[ a + d = 10 + 6 = 14 \]

Final Answer: \( a + d = 14 \)