Ratio of Radii of Concentric/Tangent Circles

CAT 2024 Slot 2 · QA · Hard · Geometry

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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

  1. A.

    2 + \sqrt{3} : 1

  2. B.

    4 + \sqrt{3} : 1

  3. C.

    7 + 4\sqrt{3} : 1

  4. D.

    4 + 2\sqrt{3} : 1

Answer

C

Explanation

Let the 3 equal circles have radius rr. Their centers form an equilateral triangle with side 2r2r. Distance from the centroid of this triangle to any vertex is R=2r3R = \frac{2r}{\sqrt{3}}.

  • Circle Y lies in the central gap, touching all three externally: rY=R−r=r(23−1)r_Y = R - r = r\left(\frac{2}{\sqrt{3}} - 1\right)
  • Circle X encloses all three circles, touching them internally: rX=R+r=r(23+1)r_X = R + r = r\left(\frac{2}{\sqrt{3}} + 1\right)

Ratio of radii: rXrY=23+123−1=2+32−3=(2+3)2=7+43\frac{r_X}{r_Y} = \frac{\frac{2}{\sqrt{3}} + 1}{\frac{2}{\sqrt{3}} - 1} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}} = (2 + \sqrt{3})^2 = 7 + 4\sqrt{3}

Ratio is 7+43:17 + 4\sqrt{3} : 1.

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