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Question

If a, b, c are the sides of a triangle ABC such that x22(a+b+c)x+3λ(ab+bc+ca)=0 has two distinct real roots, then


A

λ<43

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B

λ>53

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C

λ(43,53)

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D

λ(13,53)

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Solution

The correct option is A

λ<43


Since, roots are real, therefore D>0
4(a+b+c)212λ(ab+bc+ca)>0(a+b+c)2>3λ(ab+bc+ca)a2+b2+c2>(ab+bc+ca)(3λ2)3λ2<a2+b2+c2ab+bc+ca....(i)
Also, cosA=b2+c2a22bc1.
b2+c2a22bc
Similarly, c2+a2b22ca
and a2+b2c22ab
a2+b2+c22(ab+bc+ca)a2+b2+c2ab+bc+ca2.....(ii)
From Eqs. (i)and (ii), we get
3λ2<2λ<43


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