Parabola
Parabola
nta_pyq_2025_jan
Grade 11
Question:
Let P(4, 4\sqrt3) be a point on the parabola y 2 = 4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :
17\sqrt3 263\sqrt3
8 34\sqrt3
3 343\sqrt3
8
Step-by-Step Solution
Key Concept: Apply the core result for parabola parameters and tangents and simplify using the given constraints.
(4) 2 (4, 4\sqrt3) lies on y = 4ax \Rightarrow 48 = 4a ⋅ 4 4a = 12 \Rightarrow y 2 = 12x is equation of parabola \sqrt3 Now, parameter of P is t = 1 2 \Rightarrow Parameters of Q is t = -2 2 \Rightarrow Q( 9 4 , -3\sqrt3) \sqrt3 Area of trapezium PQNM 1 = MN ⋅ (PM + QN) 2 1 = MN ⋅ (PS + QS) 2 1 = MN ⋅ PQ 2 1 49 \sqrt3 = 7\sqrt 3 ⋅ = (343) = 3 2 4 8
Correct Answer: 4