There are 2 possible cases 1. Both centres should lie on different sides of line. 2. Line can be tangent also Centre of both the circles are (1,1) and (9,1) (3+4−α)(27+4−α)<0 (7−α)(31−α)<0 α∈(7,31)⋅⋅⋅⋅⋅⋅⋅(A) P1= Distance of line from centre (1,1) P1≥r1
|3+4−α|
5
≥1 |7−α|≥5 α∈(−∞,2]∪[12,∞)⋅⋅⋅⋅⋅⋅⋅(B) Similarly, P2= Distance of line from (9,1)
|27+4−α|
5
≥2 α∈(−∞,21]∪[41,∞)⋅⋅⋅⋅⋅⋅⋅(C) Now, (A)∩(B)∩(C)⇒α∈[12,21] Sum =(12+13+14+...+21) =(