In a ` A B C ,A-=(alpha,beta),B-=(1,2),C-=(2,3),` point `A` lies on the line `y=2x+3,` where `alpha,beta` are integers, and the area of the triangle is `S` such that `[S]=2` where `[` .`]` denotes the greatest integer function. Then the possible coordinates of `A` can be `(-7,-11)` `(-6,-9)` `(2,7)` `(3,9)`

Options

A. `(-7,-11)`
B. `(-6,-9)`
C. (2,7)
D. (3,9)

Explanation

The point `A(alpha,beta)` lies on `y=2x+3`.
Hence,
`beta=2alpha+3`
`A=(alpha, 2alpha+3)`
Area of `Delta ABC` is `:|(1)/(2)|:{:(alpha,,2alpha+3,,1),(1,,2,,1),(2,,3,,1):}:|":|`
`=|:(1)/(2)[alpha(2-3)+(2alpha+3)(2-1)+1(3-4)]:|`
`=(1)/(2)|-alpha+2alpha+3-1|=(1)/(2)|alpha+2|=S`
`[S]=2or 2leSlt3`
`therfore2le(1)/(2)|alpha+2|lt3`
`4le |alpha+2|lt6`
`|alpha+2|le6`
or `-6lealpha+2le6`
or `-8lealphale4`
and `|alpha+2|ge4`
i.e., `alpha+2ge4 or alpha+2le-4`
i.e., `alphage2 or alphale-6`
From (1) and (2) , we get
`-8ltalphalt-6or 2lealphalt4`
or `alpha=-7,-6,2,3`
The possible coordiantes of A are `(-7,-11) ,(-6,-9) ,(2,7)` and `(3,9)`.