Properties of TriangleHard
Question
Let a, b, c be the sides of a triangle. No two of them are equal and λ ∈ R. If the roots of the equation x2 + 2(a + b + c) x + 3λ (ab + bc + ca) = 0 are real, then
Options
A.λ < 
B.λ > 
C.λ ∈ 
D.λ ∈ 
Solution
D ≥ 0
⇒ 4(a + b + c)2 - 12λ (ab + bc + ca) ≥ 0
⇒ λ ≤
Since |a - b| < c ⇒ a2 + b2 - 2ab < c2 ...(1)
|b - c| < a ⇒ b2 + c2 - 2bc < a2 ...(2)
|c - a| < b ⇒ c2 + a2 - 2bc < b2 ...(3)
From (1), (2) and (3), we get
Hence λ <
⇒ λ < 
.
⇒ 4(a + b + c)2 - 12λ (ab + bc + ca) ≥ 0
⇒ λ ≤
Since |a - b| < c ⇒ a2 + b2 - 2ab < c2 ...(1)
|b - c| < a ⇒ b2 + c2 - 2bc < a2 ...(2)
|c - a| < b ⇒ c2 + a2 - 2bc < b2 ...(3)
From (1), (2) and (3), we get
Hence λ <
⇒ λ < .Create a free account to view solution
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