Straight Lines

Let R be the interior region between the lines 3x-y+1=0 and x+2y-5=0 containing the origin. The set of all values of a, for which the points (a2,a+1 ) lie in R, is :

1. (-3,0) ( 23,1 )
2. (-3,0) ( 13,1 )
3. (-3,-1) ( 13,1 )
4. (-3,-1) (- 13,1 )

Answer

(2) (-3,0) ( 13,1 )

Solution

P (a2,a+1 ) must be on the origin's side of both lines:

[figure]

L1=3x-y+1, L1(O)=1>0: 3a2-(a+1)+1>0 a(3a-1)>0 a<0 or a> 13 …(i)

L2=x+2y-5, L2(O)=-5<0: a2+2(a+1)-5<0 (a+3)(a-1)<0 -3<a<1 …(ii)

(i) ∩ (ii): a (-3,0) ( 13,1 )

Answer: (2)