If the point `(a^2,a+1)` lies in the angle between the lines `3x-y+1=0` and `x+2y-5=0` containing the origin, then find the value of `adot`

Explanation

Given liens are
`L_(1) = 3x -y+1=0 " " (1)`
`L_(2) =x+2y-5=0 " " (2)`
`P(a^(2),a+1)` lies in the angle formed by above two lines containing O(0,0). Then O and P must lie on the same side w.r.t. both the lines
` "Now ", L_(1) (0,0)= 1 gt 0`
So, we must have
`L_(1) (a^(2),a+1) gt 0`
` " or " 3a^(2)-(a+1) + 1 gt 0`
` " or " a(3a-1) gt 0`
`rArr a in (-oo, 0) uu((1)/(3), oo) " " (3)`
`L_(2)(0,0) = -5 lt 0`
So, we must have
`L_(2)(a^(2), a + 1) lt 0`
`" or " a^(2) +2(a+1)-5 lt 0`
`a^(2) +2a-3 lt 0`
`rArr (a-1) (a+3) lt 0`
` ain (-3,1) " " (4)`
From (3) and (4),
` a in (-3,0) uu((1)/(3),1)`