What is the radius of the base of the cone?

Consider the following for the next two (02) items that follow : A right conical cap just covers two spheres placed one above the other on a table such that it touches both the spheres. Let r be the radius of the smaller sphere and R be the radius of the bigger sphere. Let 2\theta be the vertical angle of the cone.

  1. A. \frac{2r^{2}\tan \theta}{R-r}
  2. B. \frac{2R^{2}\tan \theta}{R-r} ✓
  3. C. \frac{2(r^{2}+R^{2})\tan \theta}{R-r}
  4. D. \frac{(r^{2}+R^{2})\tan \theta}{R-r}

Correct Answer: B. \frac{2R^{2}\tan \theta}{R-r}

Explanation

The radius of the base of the cone is related to its total height by R_{base} = \text{height} \times \tan \theta. Substituting the height H = \frac{2R^2}{R-r}, we directly obtain R_{base} = \frac{2R^2 \tan \theta}{R-r}.

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