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Let P and Q be two points on the parabola y^2 = 8x . The tangents drawn to the parabola at P and Q intersect at a point R . A line L passing through R and parallel to the axis of the parabola meets the chord PQ at D . Let M and N be the feet of the perpendiculars drawn from P and Q respectively on the line L . If the product of the distances PM QN = 64 , then the area of the triangle PQR is equal to _________

Correct answer

128

Step-by-step solution

Let the coordinates of P and Q on the parabola y^2 = 8x be (at₁^2, 2at₁) and (at₂^2, 2at₂) respectively, where a = 8 4 = 2 . The point of intersection of the tangents at P and Q is R(at₁ t₂, a(t₁ + t₂)) . The line L passes through R and is parallel to the axis of the parabola (x-axis). Therefore, the equation of line L is y = a(t₁ + t₂) . The perpendicular distances from P and Q to the line L are PM and QN . PM = |2at₁ - a(t₁ + t₂)| = a|t₁ - t₂| QN = |2at₂ - a(t₁ + t₂)| = a|t₂ - t₁| = a|t₁ - t₂| Given that PM QN =

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