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Question

If every pair from among the equations x2 + px + qr = 0, x2 + qx + rp = 0 and x2 + rx + pq = 0 have a common root, then open parentheses fraction numerator s u m blank o f blank r o o t s over denominator p r o d u c t blank o f blank r o o t s end fraction close parentheses is

  1. fraction numerator sum p over denominator p q r end fraction    
  2. fraction numerator i over denominator p q r end fraction    
  3. (p + q + r)2    
  4. none of these.    

The correct answer is: fraction numerator sum p over denominator p q r end fraction


    Let alpha, beta be the roots of x2 + px + qr = 0…(1)
    beta, gamma be the roots of x2 + qx + rp = 0…(2)
    and gamma, alpha be the roots of x2 + rx + pq = 0…(3)
    Since beta is a common root of (1), (2)
    therefore beta to the power of 2 end exponent plus p beta plus q r equals 0 a n d beta to the power of 2 end exponent plus q beta plus r p equals 0 therefore open parentheses p minus q close parentheses beta plus open parentheses q minus p close parentheses r equals 0 rightwards double arrow beta minus r equals 0 rightwards double arrow beta equals r
    A g a i n blank alpha beta equals q r open square brackets F r o m open parentheses 1 close parentheses close square brackets rightwards double arrow alpha r equals q r rightwards double arrow alpha equals q
    Similarly from (2) and (3), we get gamma = p and from (3) and (1), beta = r
    therefore alpha plus beta plus gamma equals q plus r plus p equals p plus q plus r A g a i n open parentheses alpha beta close parentheses open parentheses beta gamma close parentheses open parentheses gamma alpha close parentheses equals open parentheses q r close parentheses open parentheses r p close parentheses open parentheses p q close parentheses rightwards double arrow open parentheses alpha beta gamma close parentheses to the power of 2 end exponent equals open parentheses p q r close parentheses to the power of 2 end exponent rightwards double arrow alpha beta gamma equals p q r therefore fraction numerator s u m o f t h e r o o t s over denominator p r o d u c t o f r o o t s end fraction equals fraction numerator alpha beta plus beta gamma plus gamma alpha over denominator alpha beta gamma end fraction equals fraction numerator p plus q plus r over denominator p q r end fraction equals fraction numerator sum p over denominator p q r end fraction

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