Find the set of all real values of λ such that the root of the equation x 2 + 2(a + b + c)x + 3λ (ab + bc + ca) = 0 are always real for any choice of a, b, c (where a, b, c represents sides of scalene triangle).
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We, know that a + b > c, b + c > a and c + a > b
⇒ c – a < b, a – b < c, b – c < a
squaring on both sides and adding (c – a) 2 + (a – b) 2 + (b – c) 2 < a 2 + b 2 + c 2
a 2 + b 2 + c 2 – 2(ab + bc + ca) < 0
⇒ (a + b + c) 2 – 4(ab + bc + ca) < 0
⇒ 
< 4....(i)
Now roots of equation x 2 + 2(a + b + c) x + 3λ (ab + bc + ca) = 0 are real, then D 
0
⇒ 4 (a + b + c) 2 – 4. 3λ (ab + bc + ca) 
0
⇒ 
3λ
So, 3λ 
< 4
⇒ λ < 
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