Let a , b , c be the sides of a triangle where a b c and . If the roots of the equation x 2 + 2 ( a + b + c ) x + 3 ( a b + b c + c a ) = 0 are real, then [2006]

Question

Let $a, b, c$ be the sides of a triangle where $a \neq b \neq c$ and $λ \in R$ . If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ are real, then [2006]

Smart Achievers · Accepted Answer

(1)

$∵$ $a, b, c are sides of a triangle and a \neq b \neq c$

$x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ $| a - b |$ $<$ $| c |$ $\Rightarrow a^{2} + b^{2} - 2 a b < c^{2} \dots (i)$

$Similarly,$

$b^{2} + c^{2} - 2 b c < a^{2} \dots (i i), c^{2} + a^{2} - 2 c a < b^{2} \dots (i i i)$

$On adding (i), (ii) and (iii) we get$

$a^{2} + b^{2} + c^{2} < 2 (a b + b c + c a)$

$\Rightarrow \frac{a^{2} + b^{2} + c^{2}}{a b + b c + c a} < 2 \dots (i v)$

$∵$ $Roots of the given equation are real$

$∴$ ${(a + b + c)}^{2} - 3 λ (a b + b c + c a) \geq 0$

$\Rightarrow \frac{a^{2} + b^{2} + c^{2}}{a b + b c + c a} \geq 3 λ - 2 \dots (v)$

$From (iv) and (v), 3 λ - 2 < 2 \Rightarrow λ < \frac{4}{3}$

Solution

Q.
Let $a, b, c$ be the sides of a triangle where $a \neq b \neq c$ and $λ \in R$ . If the roots of the equation $x^{2} + 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ are real, then [2006]

1 $λ < \frac{4}{3}$

2 $λ > \frac{5}{3}$

3 $λ \in (\frac{1}{3}, \frac{5}{3})$

4 $λ \in (\frac{4}{3}, \frac{5}{3})$

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Solution

Q. Let a,b,c be the sides of a triangle where a≠b≠c and λ∈R. If the roots of the equation x2+2(a+b+c)x+3λ(ab+bc+ca)=0 are real, then [2006]

1 λ<43

2 λ>53

3 λ∈(13,53)