Let `x^2+y^2+x y+1geqa(x+y)AAx ,y in R ,` then the number of possible integer (s) in the range of `a` is___________.

Answer

3

Explanation

`rArr (y-a)^(2)=4(y^(2)-ay+1)le0`
`rArr -3y^(2)+2ay+a^(2)-4le0AAy inR`
`therefore 3y^(2)-2ay+4-a^(2)ge0AAy in R`
`Dle0`
`rArr4a^(2)-4 " . "3(4-a^(2))le0`
`rArra^(2)-3(4-a^(2))le0rArr4a^(2)-12le0`
`therefore " Range of " a in [-sqrt3, sqrt3]`
`rArr " number of integer " {-1,0,1}`