Parabola
Parabola
nta_pyq_2025_jan
Grade 11
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 △SAB , where S is the focus of the parabola y 2 = 4x , then the value of (a + d) is ________ -
Step-by-Step Solution
Key Concept: Apply the core result for parabola parameters and tangents and simplify using the given constraints.
(14) To find image of P (t , 2t) 2 2 2 y - 2t -2 (t + 2t + 4) x - t 2 = = = -(t + 1) - 3 1 1 2 2 1 + 1 2 2 x = t - (t + 1) - 3 = -2t - 4 2 2 y = 2t - (t + 1) - 3 = -t - 4 -x - 4 t = 2 2 -x - 4 \Rightarrow y + 4 = -( ) 2 2 (x + 4) \Rightarrow (y + 4) = - 4 2 \Rightarrow x = -4y \Rightarrow Focus (-4, -5) Also, y = -5 intersect 2 \therefore (-4)(-1) = (x + 4) 2 4 = (x + 4) x + 4 = $\pm$2 x = -2, -6 \Rightarrow d = 4 ∣ 1 0 1∣ 1 ∣ ∣ a = -2 -5 1 2 ∣ ∣ ∣ -6 -5 1∣ 1 = |[1(-5 + 5) + 1(10 - 30)]| 2 1 = (20) 2 a = 10 \therefore a + d = 14
Correct Answer: 14