Concept:The chord length gives
R2−r2=144.
If both
r and
R are integers, the equation has multiple solutions.
Explanation:From the geometry, the distance from the centre to the chord is
r.
Half the chord length is
12. So
R2−r2=122=144.
Thus
(R−r)(R+r)=144.
If
r and
R are integers, then
R−r and
R+r are positive integers with same parity (both even, because their sum
2R is even).
Even factor pairs of
144 are:
(2,72),
(4,36),
(6,24),
(8,18),
(12,12).
Solving each gives possible
(R,r):
(37,35),
(20,16),
(15,9),
(13,5),
(12,0).
The pair
(12,0) is invalid because
r>0 (the chord touches the inner circle).
Thus four valid integer pairs remain.
Therefore, even using both statements together, the values of
r and
R cannot be uniquely determined.
Answer:D. The Question cannot be answered even by using both Statements together.
