CAT 2024 Slot 2 — QA Question 5
Three circles of equal radii touch (but not cross) each other externally. Two other circles, X and Y, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of X is more than that of Y, the ratio of the radii of X and Y is
Answer & solution
- A
4 + 2√3 : 1
- B
4 + √3 : 1
- C
2 + √3 : 1
7 + 4√3 : 1
Hard
Three equal circles touching pairwise have centres forming an equilateral triangle. Both X (larger, enclosing) and Y (smaller, inner) are concentric with that triangle's centroid. Get the centroid-to-centre distance, then X's radius is centroid distance and Y's is centroid distance .
Set up the geometry. Let each of the three equal circles have radius . Their centres form an equilateral triangle of side (each pair touches externally). Let be the centroid. The distance from to any centre equals the circumradius of that equilateral triangle.
Radius of the enclosing circle X. X is centred at and touches each circle externally from outside, so its radius reaches a far edge: .
Radius of the inner circle Y. Y sits in the central gap, centred at , touching each circle on its near edge: .
Take the ratio. The radius and the cancel.