There Are Only Two!

Geometry Level 3

There exist two ordered triples (a,b,c)(a,b,c) for which 4x24xy+ay2+bx+cy+1=04x^2-4xy+ay^2+bx+cy+1=0 represents a pair of identical straight lines in the xyxy-plane.

If these triples are (a1,b1,c1)(a_1,b_1,c_1) and (a2,b2,c2)(a_2,b_2,c_2), then find the value of a1+b1+c1+a2+b2+c2a_1+b_1+c_1+a_2+b_2+c_2.

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