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AP EAMCET201822 Apr 2018Morning ShiftMathematicsQuadratic EquationActual

If a, b and c are the roots of x^3+q x+r=0 , then (a-b)^2+(b-c)^2+(c-a)^2=

Options

  1. A-6 q
  2. B-4 q
  3. C6 q
  4. D4 q

Correct answer

A. -6 q

Step-by-step solution

Given, a, b and c are the roots of equation aligned & x^3+q x+r=0 , & a+b+c=0 & a b+b c+c a=q & and & a b c=-r & a+b+c=0 & (a+b+c)^2=0 & a^2+b^2+c^2+2 a b+2 b c+2 c a=0 & a^2+b^2+c^2+2(a b+b c+c a)=0 & a^2+b^2+c^2=-2 q &Now, (a-b)^2+(b-c)^2+(c-a)^2 & =a^2+b^2-2 a b+b^2+c^2-2 b c+c^2+a^2-2 ac & =2 a^2+2 b^2+2 c^2-2(a b+b c+c a) & =2 (a^2+b^2+c^2 )-2(a b+b c+c a) & =2(-2 q)-2(q) &[by Eq. (i)] & =-4 q-2 q & =-6 q . & aligned

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