Circles Part 1

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)
Sum =(12+13+14+...+21)
=(

21×22

2

)−(

11×12

2

)
=231−66=165