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Statement of a problem № 3338

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A particle describes an ellipse of eccentricity e under a force to a focus. When it approaches the nearer apse (turning point) the centre of force is transferred to the other focus. Prove that the eccentricity of the new orbit is e(3 + e)/(1-e).

Solution:
A particle describes an ellipse of eccentricity e under a force to a focus. When

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