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Question

text  If  end text f left parenthesis x right parenthesis equals fraction numerator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis over denominator x end fraction text  and  end text fraction numerator f left parenthesis x right parenthesis over denominator left parenthesis x minus y right parenthesis left parenthesis x minus z right parenthesis end fraction plus fraction numerator f left parenthesis y right parenthesis over denominator left parenthesis y minus z right parenthesis left parenthesis y minus x right parenthesis end fraction plus fraction numerator f left parenthesis z right parenthesis over denominator left parenthesis z minus x right parenthesis left parenthesis z minus y right parenthesis end fraction equals fraction numerator K over denominator x y z end fraction text , then  end text K equals

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

Hint:

When x = a ,f(a) = 0 or when x = b , f(b) = 0
Substitute the respective values if f(z) and f(y) by taking x as z and y in the equation and find the answer.

The correct answer is: a b


    text end text f left parenthesis x right parenthesis equals fraction numerator left parenthesis x minus a right parenthesis left parenthesis x minus b right parenthesis over denominator x end fraction left parenthesis 1 right parenthesis space space space space space space text  and  end text
fraction numerator f left parenthesis x right parenthesis over denominator left parenthesis x minus y right parenthesis left parenthesis x minus z right parenthesis end fraction plus fraction numerator f left parenthesis y right parenthesis over denominator left parenthesis y minus z right parenthesis left parenthesis y minus x right parenthesis end fraction plus fraction numerator f left parenthesis z right parenthesis over denominator left parenthesis z minus x right parenthesis left parenthesis z minus y right parenthesis end fraction equals fraction numerator K over denominator x y z end fraction left parenthesis 2 right parenthesis
    When x = a , Substitute the value of x in equation 1
    f left parenthesis a right parenthesis space equals space fraction numerator left parenthesis a minus a right parenthesis left parenthesis a minus b right parenthesis over denominator a end fraction space equals space 0
    f(y) = fraction numerator left parenthesis y minus a right parenthesis left parenthesis y minus b right parenthesis over denominator y end fraction and f (z) = fraction numerator left parenthesis z minus a right parenthesis left parenthesis z minus b right parenthesis over denominator z end fraction
    Substitute these values in equation 2 we get
    
rightwards double arrow fraction numerator f left parenthesis x right parenthesis over denominator left parenthesis x minus y right parenthesis left parenthesis x minus z right parenthesis end fraction plus fraction numerator f left parenthesis y right parenthesis over denominator left parenthesis y minus z right parenthesis left parenthesis y minus x right parenthesis end fraction plus fraction numerator f left parenthesis z right parenthesis over denominator left parenthesis z minus x right parenthesis left parenthesis z minus y right parenthesis end fraction equals fraction numerator K over denominator x y z end fraction
    rightwards double arrow 0 space plus space fraction numerator left parenthesis y minus a right parenthesis left parenthesis y minus b right parenthesis over denominator y space left parenthesis y minus z right parenthesis left parenthesis y minus a right parenthesis end fraction space plus space fraction numerator left parenthesis z minus a right parenthesis left parenthesis z minus b right parenthesis over denominator z space left parenthesis z minus a right parenthesis left parenthesis z minus y right parenthesis end fraction equals space fraction numerator K over denominator a y z end fraction
    rightwards double arrow fraction numerator left parenthesis y minus b right parenthesis over denominator y left parenthesis y minus z right parenthesis end fraction space plus space fraction numerator left parenthesis z minus b right parenthesis over denominator z left parenthesis z minus y right parenthesis end fraction equals space fraction numerator K over denominator a y z end fraction
    rightwards double arrow fraction numerator y z space minus space b z minus y z space plus space y b over denominator y z left parenthesis y minus z right parenthesis end fraction space equals space fraction numerator K over denominator a y z end fraction
rightwards double arrow fraction numerator b left parenthesis y minus z right parenthesis over denominator left parenthesis y minus z right parenthesis end fraction space equals space K over a
rightwards double arrow K space equals space a b

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