Maths-
General
Easy

Question

If ell1 & ell2 are length of segments of focal chord of parabola y2 = 4ax than harmonic mean of ell1 & ell2 is equal to

  1. 4a    
  2. 3a    
  3. 2a    
  4. a    

The correct answer is: 2a


    To find the harmonic mean of the segments of a focal chord.
    Coordinates of focal chord
    A=open parentheses a t subscript 1 squared comma 2 a t subscript 1 close parentheses
    B=open parentheses a t subscript 2 squared comma 2 a t subscript 2 close parentheses
    Harmonic mean = fraction numerator 2 over denominator begin display style 1 over A end style plus begin display style 1 over B end style end fraction equals fraction numerator 2 over denominator begin display style fraction numerator 1 over denominator a t subscript 1 squared plus a end fraction end style plus begin display style fraction numerator 1 over denominator a t subscript 2 squared plus a end fraction end style end fraction space equals space fraction numerator 2 a over denominator fraction numerator 1 over denominator t subscript 1 squared plus 1 end fraction plus fraction numerator 1 over denominator t subscript 2 squared plus 1 end fraction end fraction
    t subscript 2 equals negative 1 over t subscript 1

rightwards double arrow fraction numerator 2 a over denominator begin display style fraction numerator 1 plus t squared over denominator t squared plus 1 end fraction end style end fraction space equals space 2 a

    Therefore, harmonic mean of the segments of a focal chord is 2a

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