L1: ax+2y+8=0, L2: 4x+(a+2)y-6=0, L3: y=0
If a=-2, three lines can be rewritten: L1: y=x-4, L2: x=3/2, L2: y=0 These 3 lines have different slopes, and form a triangle. Therefore, a≠0 and three lines can be rewritten as follows:
L1: y=-ax/2 - 4, L2: y=-4x/(a+2) + 6/(a+2), L3: y=0
Since a≠0 and a≠-2, L1 and L2 are not parallel to L3. Thus, L1 and L2 must be parallel to each other not to form a triangle. So, the slopes of L1 and L2 must be identical.
That is: -a/2=-4/(a+2) Simplify the equation, getting (a+4)(a-2)=0 We have: a=-4, or a=2
CK: Plug the values of a into the equations of L1 and L2.
If a=-4, L1: y=2x - 4, L2: y=2x - 3/2, so L1 and L2 are parallel to each other.
If a=2, L1: y=-x - 4, L2: y=-x + 3/2, so L1 and L2 are parallel to each other. CKD.
Therefore, the answer is: a=-4, or a=2