Official solution
S : 36x2 + 36y2 - 108x + 120y + C = 0
\[\Rightarrow\] x2 + y2 - 3x + \[{{10} \over 3}\]y + \[{C \over {36}}\] = 0
Centre \[ \equiv ( - g, - f) \equiv \left( {{3 \over 2},{{ - 10} \over 6}} \right)\]
radius = \[r = \sqrt {{9 \over 4} + {{100} \over {36}} - {C \over {36}}} \]
Now,
\[ \Rightarrow r < {3 \over 2}\]
\[ \Rightarrow {9 \over 4} + {{100} \over {36}} - {C \over {36}} < {9 \over 4}\]
\[\Rightarrow\] C > 100 ...... (1)
Now, point of intersection of x - 2y = 4 and 2x - y = 5 is (2, -1), which lies inside the circle S.
\[\therefore\] S(2, -1) < 0
\[\Rightarrow\] (2)2 + (-1)2 - 3(2) + \[{{10} \over 3}(-1) + \[{C \over {36}}\] < 0
\[\Rightarrow\] 4 + 1 - 6 - \[{{10} \over 3}\] + \[{C \over {36}}\] < 0
C < 156 ..... (2)
From (1) \& (2)
100 < C < 156 Ans.