Proof That The Curve Has Constant Sum of Distances To Foci
Consider the illustration, depicting a plane intersecting a cone to form an ellipse (the interior of the ellipse is colored light blue). The two Dandelin spheres are shown, one (G1) above the ellipse, and one (G2) below. The intersection of each sphere with the cone is a circle (colored white).
- Each sphere touches the plane at a point, and let us call those two points F1 and F2.
- Let P be a typical point on the ellipse.
- Prove: The sum of distances d(F1, P) + d(F2, P) 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 ellipse, 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 lines PF1 and PP1 are both tangent to the same sphere (G1).
- Likewise, the distance from F2 to P is the same as the distance from P2 to P, because lines PF2 and PP2 are both tangent to the same sphere (G2).
- Consequently, the sum of distances d(F1, P) + d(F2, P) must be constant as P moves along the curve because the sum of distances d(P1, P) + d(P2, P) also remains constant.
- 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.
This proves a result that had been proved in a different manner by Apollonius of Perga.
If (as is often done) one takes the definition of the ellipse to be the locus of points P such that d(F1, P) + d(F2, P) = a constant, then the argument above proves that the intersection of a plane with a cone is indeed an ellipse. That the intersection of the plane with the cone is symmetric about the perpendicular bisector of the line through F1 and F2 may be counterintuitive, but this argument makes it clear.
Adaptations of this argument work for hyperbolas and parabolas as intersections of a plane with a cone. Another adaptation works for an ellipse realized as the intersection of a plane with a right circular cylinder.
Read more about this topic: Dandelin Spheres
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