CAT 2019 Slot 2 — QA Question 7
Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is
Answer & solution
Correct answer: 1
- A
π/3
1
- C
1/√2
- D
√2
Easy
All three circles sit on the same straight tangent line, so each centre is at a height equal to its radius. Connect the centre of one big circle (radius ) to the small circle (radius ); the centre-to-centre distance is (external touch) and the horizontal/vertical legs come from the radii. Apply the Pythagorean theorem.
Set up the right triangle. Let the small circle have radius . Each centre lies above the common tangent at a height equal to its radius. Taking a big-circle centre (height ) and the small-circle centre (height ), the right triangle on the line has:
Find the horizontal leg. The two equal circles touch externally, so the foot of the big circle on the tangent is from the contact point — the horizontal distance from to works out to . Apply the Pythagorean theorem in .
Solve for . Expand both squares from step 2.