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Let a,b,c be the sides of a triangle. No...

Let `a,b,c` be the sides of a triangle. No two of them are equal and `lambda in R` If the roots of the equation `x^2+2(a+b+c)x+3lambda(ab+bc+ca)=0` are real, then

A

`lamda lt 4/3`

B

`lamda lt 5/3.`

C

`lamda epsilon(1/3,5/3)`

D

`lamda epsilon (4/3,5/3)`

Text Solution

Verified by Experts

The correct Answer is:
A
`Dge0`
`4(a+b+c)^(2)-12lamda(ab+bc+ca)ge0`
`(a^(2)+b^(2)+c^(2))-(3lamda-2)(ab+bc+ca)ge0`
`:.(3lamda-2)le((a^(2)+b^(2)+c^(2)))/((ab+bc+ca))`
Since `|a-b|ltc`
`impliesa^(2)+b^(2)-2abltc^(2)`……i
`|b-c|lta`
`impliesb^(2)+c^(2)-2bclta^(2)`.......ii
`|c-a|ltb`
`impliesc^(2)+a^(2)-2caltb^(2)`..........iii
From Eqs i, ii and iii we get
`(a^(2)+b^(2)+c^(2))/(ab+bc+ca)lt2`............iv
FromEqs i and iv we get
`3lamda-2lt2implieslamdalt4/3`
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