Let `A(0,beta), B(-2,0)` and `C(1,1)` be the vertices of a triangle. Then
Angle A of the triangle ABC will be obtuse if `beta` lies in

Options

A. `(-1,2)`
B. `(2,(5)/(2))`
C. `(-1,(2)/(3))uu((2)/(3),2)`
D. none of these

Answer

C

Explanation

Since `cos A = (AB^(2)+AC^(2)-BC^(2))/(2.AB.AC)`
A is obtuse
`:' cos A lt 0`
`:. AB^(2)+AC^(2) lt BC^(2)`
Also, A,B,C and non-collinear.
`rArr 10 gt beta^(2) +4 +1 +(beta -1)^(2)`
`rArr (beta -2) (beta +1) lt 0`
`rArr beta in (-1,2)`
But A,B,C are collinear for `beta = (2)/(3)`
`rArr` correct interval for B is
`(-1,(2)/(3))uu((2)/(3),2)`