Let P(4, 4√3) be a point on the parabola y² = 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:

(1) (263√3)/8 

(2) 17√3 

(3) (343√3)/8 

(4) (34√3)/3

Point P(4, 4√3) lies on y² = 4ax

(4√3)² = 4a(4) → 48 = 16a → a = 3

Parabola equation: y² = 12x

For point P, using x = at², y = 2at:

• 4 = 3t₁² → t₁ = 2/√3

For focal chord: t₂ = -1/t₁ = -√3/2

Point Q: x = 3 × 3/4 = 9/4, y = 6 × (-√3/2) = -3√3

So Q(9/4, -3√3)

For trapezium PQNM (where M, N are feet of perpendiculars on directrix):

Distance MN represents the vertical distance, and PM + QN equals the focal chord PQ

Area of trapezium = (1/2) × MN × PQ

Using focal chord formula with parameters t₁ and t₂:

Area = (1/2) × (49/4) × (7√3/2) = (343√3)/8

Answer: (3) (343√3)/8