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) `lambda < 4/3` (b) `lambda > 5/3` (c) `lambda in (1/5,5/3)` (d) `lambda in (4/3,5/3)`

Options

A. `lamda lt 4/3`
B. `lamda lt 5/3.`
C. `l epsilon(1/3,5/3)`
D. `lamda epsilon (4/3,5/3)`

Answer

A

Explanation

`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`