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Let P be a point in the first quadrant on the parabola y^2 = 8x , 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 directrix of the parabola, and the area of the quadrilateral PQNM is 108 , then the length of the focal chord PQ is equal to :

Options

  1. A6
  2. B9
  3. C18
  4. D36

Correct answer

C. 18

Step-by-step solution

The equation of the parabola is y^2 = 8x , so a = 2 . The directrix is x = -2 . Let the parameter of point P be t ( t > 0 ). Then P is (2t^2, 4t) . Since PQ is a focal chord, the parameter of Q is - 1 t , so Q is ( 2 t^2 , - 4 t ) . The feet of the perpendiculars on the directrix are M(-2, 4t) and N (-2, - 4 t ) . The quadrilateral PQNM is a trapezium with parallel sides PM and QN . The lengths of the parallel sides are: PM = 2t^2 - (-2) = 2(t^2 + 1) QN = 2 t^2 - (-2) = 2 ( 1 t^2 + 1 ) The distance between the para

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