A and B are centres of two circles with radii 2 cm and 1 cm respectively, where AB = 5 cm. C is the centre of another circle of radius r cm, which touches each of the above two circles externally. If ∠ACB=90° , then the value of r is:
Show answer & explanation
Correct answer: B
Explanation
The correct answer is option 2 i.e. 2 cm
Circle 1: Center A, radius 2 cm
Circle 2: Center B, radius 1 cm
Distance AB: 5 cm
Circle 3: Center C, radius r cm, touches both circles externally.
The distance between A and C = 2 + r
The distance between B and C = 1 + r
Given that ∠ACB=90º, the centers A, C, and B form a right triangle:
⇒ AC2 + BC2 = AB2
⇒ (2 + r)2 + (1 + r)2 = 52
⇒ 4 + 4r + r2 + 1 + 2r + r2 = 25
⇒ 2r2 + 6r + 5 = 25
⇒ 2r2 + 6r − 20 = 0
⇒ r2 + 3r − 10 = 0
⇒ (r + 5)(r − 2) = 0
Thus:
⇒ r = −5 or r = 2
Since r > 0, we have:
r = 2
The radius of the third circle = 2 cm