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Prove that similar ellipses have the same eccentricity

  1. Prove that a similarity transform carries an ellipse with center at the origin to another ellipse with the same eccentricity.
  2. Prove the converse of part (a), i.e., two concentric ellipses with the same eccentricity and major axes on the same line are related by a similarity transform.
  3. Prove the statements corresponding to parts (a) and (b) for hyperbolas.