Let P(16, 16) be a point on the parabola y^2 = 16x and PQ be a focal chord of the parabola. If M and N are the feet of the perpendiculars drawn from P and Q respectively on the tangent at the vertex of the parabola, then the area of the quadrilateral PQNM is equal to :

Step-by-step solution

For the parabola y^2 = 16x , we have 4a = 16 a = 4 . The tangent at the vertex is the y-axis ( x = 0 ). Let the parameter of point P(16, 16) be t₁ . Comparing with (at₁^2, 2at₁) , we get 2(4)t₁ = 16 t₁ = 2 . Since PQ is a focal chord, the parameter of Q is t₂ = - 1 t₁ = - 1 2 . The coordinates of Q are (at₂^2, 2at₂) = (4 (- 1 2 )^2, 2(4) (- 1 2 ) ) = (1, -4) . The feet of the perpendiculars from P and Q on the y-axis are M(0, 16) and N(0, -4) respectively. The quadrilateral PQNM is a trapezium with parallel sides P