# 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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