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AP EAMCET201825 Apr 2018Morning ShiftMathematicsQuadratic EquationActual

Let a , b and c be three positive real numbers such that the sum of any two of them is greater than the third. All the values of λ such that the roots of the equation x 2 + 2 ( a + b + c ) x + 3 λ ( a b + b c + c a ) = 0 are real, are given by

Options

  1. Aλ < 2 3
  2. Bλ ≥ 2 3
  3. Cλ < 4 3
  4. D1 3 < λ < 2 3

Correct answer

C. λ < 4 3

Step-by-step solution

Given, x 2 + 2 ( a + b + c ) x + 3 λ ( a b + b c + c a ) = 0 For the roots to be real, D ≥ 0 4 ( a + b + c ) 2 - 12 λ a b + b c + c a ≥ 0 ⇒ ( a + b + c ) 2 - 3 λ a b + b c + c a ≥ 0 ⇒ a 2 + b 2 + c 2 + 2 ∑ a b ≥ 3 λ ∑ a b ⇒ 1 3 ∑ a 2 ∑ a b + 2 3 ≥ λ       . . . 1 Now, it is given that a + b > c       . . . i b + c > a       . . . i i c + a > b      

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