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Ellipse Proof: The sum of distances d(F1P) + d(F2P) remain constant as the point P moves along the curve.

  • A line passing through P and the vertex S of the cone intersects the two circles at points P1 and P2.
  • As P moves along the curve, P1 and P2 move along the two circles.
  • The distance from F1 to P is the same as the distance from P1 to P, because these distances are the lengths of the line segments PF1 and PP1, which have a common endpoint P and are both tangent to the same sphere (G1) at their other endpoints.
  • Likewise, by a symmetrical argument, the distance from F2 to P is the same as the distance from P2 to P.
  • Consequently, the sum of distances d(F1P) + d(F2P) must be constant as P moves along the curve because the sum of distances d(P1P) + d(P2P) also remains constant. Therefore, the curve is an ellipse.  This follows from the fact that P lies on the straight line from P1 to P2, and the distance from P1 to P2 remains constant.

 

Categories: Math

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