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AP EAMCET2010MathematicsQuadratic Equation

The condition that the roots of x^3-b x^2+c x-d=0 are in geometric progression is

Options

  1. Ac^3=b^3 d
  2. Bc^2=b^2 d
  3. Cc=b d^3
  4. Dc=b d^2

Correct answer

A. c^3=b^3 d

Step-by-step solution

Equation, x^3-b x^2+c x-d=0 Let the roots of this cubic equation in GP are ( a r , a, a r ) Then, Sum of the roots a r +a+a r=- ( -b 1 ) b a ( 1 r +1+r )=b (i) and a r a+a a r+a r a r = c 1 a^2 ( 1 r +r+1 )=c (ii) Product of the roots a r a a r=- ( -d 1 )=d a^3=d (iii) Eq. (ii) divided by Eq. (i), we get a=c / b , put the value of ' a ' in Eq. (iii) (c / b)^3=d c^3=b^3 d

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