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Let A and B be two points on the parabola y^2 = 10x such that the chord AB subtends a right angle at the vertex V of the parabola. Let the chord AB intersect the axis of the parabola at point D . If M and N are the feet of the perpendiculars drawn from A and B respectively on the axis of the parabola, then the value of VM VN + AM BN VD is equal to _________

Correct answer

20

Step-by-step solution

Let the coordinates of points A and B on the parabola y^2 = 10x be (at₁^2, 2at₁) and (at₂^2, 2at₂) respectively, where a = 10 4 = 2.5 . The vertex of the parabola is V(0,0) . Since the chord AB subtends a right angle at the vertex V , the product of the slopes of VA and VB is -1 . 2at₁ at₁^2 2at₂ at₂^2 = -1 4 t₁ t₂ = -1 t₁ t₂ = -4 The points M and N are the feet of the perpendiculars from A and B on the axis of the parabola (x-axis). Thus, M = (at₁^2, 0) and N = (at₂^2, 0) . The distances are VM = at₁^2 and VN = at

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