Thursday, April 8, 2010

Fox 269 - Solution

By Yu:

0 ≤ α ≤ 60
Large Ellipse
Semi-major axis: (1/2)(√3 + cot(α/2))
Semi-minor axis: (1/2)(√3 - tan(α/2))
Area = π*major*minor=(π/2)(√3 cotα + 1).

Small Ellipse

Semi-major axis: (1/2)(tan(α/2) + √3)
Semi-minor axis: (1/2)(cot(α/2) - √3)
Area = π*major*minor = (π/2)(√3 cotα - 1).

Ratio of their areas (large : small) = (√3 cotα + 1)/(√3 cotα - 1)
= [(√3/2)cosα + (1/2)sinα] / [(√3/2)cosα - (1/2)sinα]
= sin(60+α) / sin(60-α)

60 ≤ α ≤ 90
Large Ellipse: Area = (π/2)(1 + √3 cotα)

Small Ellipse: Area = (π/2)(1 - √3 cotα)
Ratio of their areas (large : small) = (1 + √3 cotα)/(1 - √3 cotα)
= [(1/2)sinα + (√3/2)cosα] / [(1/2)sinα - (√3/2)cosα]
= sin(α+60) / sin(α-60).
An observation:
When α=60, the ratio is undefined because the area of the small ellipse is 0. The third vertex maps out a straight line.

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