Rotating ellipse

Geometry Level 5

An ellipse \(E\) is rotating about its center with a constant angular speed \(\omega \) in the counter clockwise direction. However, during its rotation, its eccentricity, \(e\), is varying with time so as to satisfy the following criteria.

  1. The foci of the ellipse \(E\) trace out another fixed, concentric ellipse \(E'\) whose eccentricity is \(\displaystyle \frac{1}{2} \).

  2. The foci of the ellipse \(E'\) always lie on the ellipse \(E\).

Evaluate the root mean square eccentricity of the ellipse \(E\) i.e., \(\displaystyle \sqrt{< e^2 > } \).

Details and Assumptions:

  • Both the ellipses are concentric as mentioned before. Moreover, the length of semi-major axis of both the ellipses is the same equal to \( a = 5 \).

  • The conditions are satisfied at all instants of time and hence the eccentricity is continuous function of time except when their axes are parallel.

  • The average has to be taken over one complete rotation.

  • The root mean square of continuous function \( p(x) \) defined as \(\displaystyle [ a , b ] \to \mathbb{R} \) is given as \( \displaystyle \sqrt{< (p(x))^2 > } = \sqrt{\frac{\int \limits_a^b (p(x))^2 \text{ d}x }{\int \limits_a^b \text{ d}x }} \)


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