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General
Easy

Question

If the tangent at the point P(x1, y1) to the parabola y2 = 4ax meets the parabola y2 = 4a(x + b) at Q and R, then the mid-point of QR is -

  1. (x1 + b, y1 + b)    
  2. (x1 – b, y1 – b)    
  3. (x1, y1)    
  4. (x1 + b, y1)    

The correct answer is: (x1, y1)


    Equation of the tangent at P(x1, y1) to y2 = 4ax is
    yy1 – 2ax – 2ax1 = 0....(1)
    Equation of the chord of y2 = 4a(x + b) whose mid-point is (x', y') is
    yy' – 2ax – 2ax' – 4ab = y'2 – 4ax' – 4ab
    (i.e.) yy' – 2ax – (y'2 – 2ax') = 0
    Equations (1) and (2) represent the same line
    therefore fraction numerator y subscript 1 end subscript over denominator y ´ end fraction equals fraction numerator 2 a over denominator 2 a end fraction equals fraction numerator 2 a x subscript 1 end subscript over denominator y ´ to the power of 2 end exponent – 2 a x ´ end fraction
    This gives y' = y1 and then 2ax1 = y'2 – 2ax'
    = y12 – 2ax'
    = 4ax1 – 2ax'
    therefore x' = x1
    therefore mid-point (x', y') = (x1, y1).

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