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

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 solution | Comment 2 of 3 |
The area of an ellipse equals pi*a*b, where a and b are the major and minor semi-axes. Maximizing a*b will maximize pi*a*b.

If x is half the distance between the centers of the circles,

b^2 + x^2 = 1

a = 1 + x

b = sqrt(1 - x^2)

a*b = (1 - x^2)^(1/2) * (1 + x)

Its derivative is

(1 - x^2)^(1/2) + (1 + x) * ((1 - x^2)^(-1/2) / 2) * (-2*x)

Wolfram Alpha finds x = 1/2 is where this derivative is zero.

The major semi-axis is 3/2 and the minor semi-axis is sqrt(3/4) = sqrt(3)/2.

As a check:

Wolfram Alpha also is asked to maximize (1 - x^2)^(1/2) * (1 + x):

max{sqrt(1 - x^2) (1 + x)} = (3 sqrt(3))/4 at x = 1/2

The area of the ellipse is pi * (3 sqrt(3))/4

 Posted by Charlie on 2024-02-21 08:10:04

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