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Ellipse covered by 2 circles (Posted on 2024-02-21) Difficulty: 2 of 5
Find the ellipse of maximum area that can be completely covered by two unit circles.

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Solution Solution | Comment 1 of 3
The ellipse (x/a)^2 + (y/b)^2 = 1 intersects the axes at:
(±a,0) and (0,±b).
Its area is π*ab.

If the circles are:
(x-t)^2 + y^2 = 1 and
(x+t)^2 + y^2 = 1
The outer halves of each circle intersect the axes at:
(±(t+1),0) and (0,±√(1-t^2))

So let a = t+1
       b = √(1-t^2)

area A = pi*(t+1)*√(1-t^2)  (to be maximized)

A' = π(√(1-t^2) + (t+1)*(1-t^2)^(-1/2)*(1/2)(-2t)
Set to zero, divide by π, multiply by √(1-t^2)
(1-t^2) = (t+1)t
2t^2 + t - 1 = 0
(2t-1)(t+1) = 0
t = {1/2, -1}

t = 1/2; a=3/2; b = √3/2
Area = π*3√3/4 = approx 4.08104856953

diagram on Desmos:
https://www.desmos.com/calculator/6sy5fchucj

  Posted by Larry on 2024-02-21 07:26:28
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