Let `A(0,beta), B(-2,0)` and `C(1,1)` be the vertices of a triangle. Then
All the values of `beta` for which angle A of triangle ABC is largest lie in the interval

Options

A. `(-2,1)`
B. `(-2,(2)/(3))uu((2)/(3),1)`
C. `(-2,(2)/(3))uu((2)/(3),sqrt(6))`
D. none of these

Answer

C

Explanation

Whenever A is obtuse is the largest angle also. But there many be more values of `beta` for which A is largest but is not obtuse. Angle A will be largest if `BC gt AC, BC gt AB, A,B,C` are not collinear
`rArr 10 gt 1 +(beta -1)^(2),10 gt 4 +beta^(2), beta ne (2)/(3)`
`rArr beta^(2)-2beta -8 lt 0, beta^(2) lt 6`
`rArr-2 lt beta lt 4, -sqrt(6) lt beta sqrt(6)`
`beta in (-2,(2)/(3))uu((2)/(3),sqrt(6))`