SATHEE: Theory of Equations 1 Question 23

Theory of Equations 1 Question 23

24. If $a, b, c$ are the sides of a triangle $A B C$ such that $x^{2} - 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ has real roots, then

$(2006, 3 M)$

(a) $λ < \frac{4}{3}$

(b) $λ > \frac{5}{3}$

(d) $λ \in \frac{1}{3}, \frac{5}{3}$

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Answer:

Correct Answer: 24. (a)

Solution:

Since, roots are real, therefore $D \geq 0$

$\begin{array}{lc} \Rightarrow & 4 (a + b + c)^{2} - 12 λ (a b + b c + c a) \geq 0 \\ \Rightarrow & (a + b + c)^{2} \geq 3 λ (a b + b c + c a) \\ \Rightarrow & a^{2} + b^{2} + c^{2} \geq (a b + b c + c a) (3 λ - 2) \\ \Rightarrow & 3 λ - 2 \leq \frac{a^{2} + b^{2} + c^{2}}{a b + b c + c a} \\ Also, & \cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c} < 1 \end{array}$

$\begin{array}{ll} \Rightarrow & b^{2} + c^{2} - a^{2} < 2 b c \\ Similarly, & c^{2} + a^{2} - b^{2} < 2 c a \\ and & a^{2} + b^{2} - c^{2} < 2 a b \\ \Rightarrow & a^{2} + b^{2} + c^{2} < 2 (a b + b c + c a) \\ \Rightarrow & \frac{a^{2} + b^{2} + c^{2}}{a b + b c + c a} < 2 \end{array}$

From Eqs. (i) and (ii), we get

$3 λ - 2 < 2 \Rightarrow λ < \frac{4}{3}$

Theory of Equations 1 Question 23

24. If $a, b, c$ are the sides of a triangle $A B C$ such that $x^{2} - 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ has real roots, then

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Theory of Equations 1 Question 23

24. If a,b,c are the sides of a triangle ABC such that x2−2(a+b+c)x+3λ(ab+bc+ca)=0 has real roots, then

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24. If $a, b, c$ are the sides of a triangle $A B C$ such that $x^{2} - 2 (a + b + c) x + 3 λ (a b + b c + c a) = 0$ has real roots, then