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Let A B C be a triangle such that /A C B...

Let `A B C` be a triangle such that `/_A C B=pi/6` and let `a , ba n dc` denote the lengths of the side opposite to `A , B ,a n dC` respectively. The value(s) of `x` for which `a=x^2+x+1,b=x^2-1,a n dc=2x+1` is(are)

A

`-(2+sqrt(3))`

B

`1+sqrt(3)`

C

`2+sqrt(3)`

D

`4sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
Using , `cosC=(a^(2)+b^(2)-c^(2))/(2ab)`

`rArr(sqrt(3))/(2)=((x^(2)+x+1)^(2)+(x^(2)-1)^(2)-(2x+1)^(2))/(2(x^(2)+x+1)(x^(2)-1))`
`rArr(x+2)(x+1)(x-1)x+(x^(2)-1)^(2)=sqrt(3)(x^(2)+x+1)(x^(2)-1)`
`rArrx^(2)+2x+(x^(2)-1)=sqrt(3)(x^(2)+x+1)`
`rArr(2-sqrt(3))x^(2)+(2-sqrt(3))x-(sqrt(3)+1)=0`
`rArrx=-(2-sqrt(3))andx=1+sqrt(3)`
But , `x=-(2+sqrt(3))rArrc` is negative ,which is not possible `x=1+sqrt(3)` is the only solution.
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