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AP EAMCET201920 Apr 2019Morning ShiftMathematicsQuadratic EquationActual

Let (a, b ) and (c ) be the sides of a scalane triangle. If ( ) is a real number such that the roots of the equation (x^2+2(a+b+c) x+3 (a b+b c+c a)=0 ) are real, then the interval in which ( ) lies is

Options

  1. A( (- , 4 3 ) )
  2. B( ( 5 3 , ) )
  3. C( ( 1 3 , 5 3 ) )
  4. D( ( 4 3 , ) )

Correct answer

A. ( (- , 4 3 ) )

Step-by-step solution

It is given that roots of given quadratic equation (x^2+2(a+b+c) x+3 (a b+b c+c a)=0 ) are real, so ( gathered D 0 4(a+b+c)^2-4 3 (a b+b c+c a) 0 (a+b+c)^2-3 (a b+b c+c a) 0 (a+b+c)^2 3(a b+b c+c a) gathered ) Now, for scalane triangle. ( aligned & For A B C|b-c| < a,|c-a| < b and |a-b| < c & (b-c)^2+(c-a)^2+(a-b)^2 < a^2+b^2+c^2 & a^2+b^2+c^2 < 2(a b+b c+c a) & a^2+b^2+c^2+2(a b+b c+c a) < 4(a b+b c+c a) & (a+b+c)^2 3(a b+b c+c a) < 4 3 & < 4 3 aligned ) Hence, option (1) is correct.

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