Does the Axis of a Cone Cross Through a Focus of an Ellipse?

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    Cone Ellipse Focus
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Discussion Overview

The discussion revolves around the geometric relationship between the axis of a cone and the foci of an ellipse formed as a conic section. Participants explore whether the axis of the cone intersects with one of the foci of the ellipse and seek mathematical justification for this relationship.

Discussion Character

  • Exploratory, Conceptual clarification

Main Points Raised

  • One participant questions if the axis of a cone intersects with one of the foci of an ellipse constructed from that cone.
  • Another participant asserts that the foci are defined by the points where Dandelin spheres touch the plane of the ellipse, suggesting that the axis does not intersect with the foci.
  • A follow-up inquiry asks if the axis of the cone has any additional geometric significance beyond its relationship to the foci.
  • A later reply indicates uncertainty about any further geometric significance of the cone's axis.

Areas of Agreement / Disagreement

Participants do not reach a consensus; while one participant asserts that the axis does not intersect the foci, the discussion remains open regarding the geometric significance of the cone's axis.

TheFerruccio
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An ellipse is a conic section. If you construct an ellipse using a cone, does the axis of the cone cross through one of the foci of the ellipse?

if so, how can this be shown mathematically?

This is just purely out of curiosity.
 
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Hi TheFerruccio! :smile:
TheFerruccio said:
If you construct an ellipse using a cone, does the axis of the cone cross through one of the foci of the ellipse?

Sorry … noooo.

The foci are the points at which the Dandelin spheres touch the plane of the ellipse (or hyperbola) … see the PF library entry " conic " for details. :smile:
 
that makes sense. So, does the axis of the cone have any geometric significance in this case beyond just what was described?
 
TheFerruccio said:
that makes sense. So, does the axis of the cone have any geometric significance in this case beyond just what was described?

Not that I know of … :blushing:
 

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