2023 · Q39
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 2θ be the vertical angle of the cone. What is the height of the cone?
Elementary MathematicsTrigonometry
- A2r²/(R − r)
- B2R²/(R − r)
- C2(r² + R²)/(R − r)
- D(r² + R²)/(R − r)
Explanation
Let the cone's semi-vertical angle be θ. The perpendicular from each sphere's centre to the slant side equals its radius, so with apex A and centres C₁ (big), C₂ (small): AC₁·sinθ = R and AC₂·sinθ = r. The spheres touch, so C₁C₂ = R + r = AC₁ − AC₂ = (R − r)/sinθ, giving sinθ = (R − r)/(R + r). The cone's height is the apex-to-table distance: H = AC₁ + R = R/sinθ + R = R(R + r)/(R − r) + R = 2R²/(R − r). The answer is (B).