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The theorem: Let AB be a focal chord of a parabola, let C be the circle with diameter AB, let V be the vertex of the parabola. The power of V with respect to C independent of the choice of the focal chord AB.

It is relatively simple to prove this using coordinate geometry. My question is: is there a simple and/or intuitive reason for this property that doesn’t require a coordinate geo bash? (e.g. reasoning using the definition of a parabola)

Isaac Greene
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  • In my book, there is only one focal chord (it is parallel to the directrix and orthogonal to the axis of symmetry). You seem to have a somewhat freer definition. It is any chord passing through the focal point? – Arthur Jul 06 '18 at 06:51
  • This answer incorporates a proof of the power property as part of a related result. It's possible that the questions are related enough for this to qualify as a duplicate. – Blue Jul 06 '18 at 12:23
  • @Arthur Yes that is correct. I would consider any chord passing through the focus as a focal chord; and I would call the focal chord parallel to the directrix the "latus rectum" – Isaac Greene Jul 06 '18 at 12:33
  • @Blue It turns out that the post you linked answers the exact question I was trying to solve! Thankyou! How did you learn to apply euclidean geometry to conic sections? Could you recommend any resources? – Isaac Greene Jul 06 '18 at 12:39
  • @IsaacGreene: If the other question satisfies you, then you should close this one. As for learning the geometry of conic sections, you could/should ask a "reference request" question for recommendations. ... I really only "know" the basics: focus definitions, focus-directrix definitions, Dandelin spheres, reflection properties, other bits and pieces I've picked up over the years. To be honest, I often attack conic section problems with coordinates and trig to glean information about various relations (like the constant power in your problem), then work to devise geometric solutions. – Blue Jul 08 '18 at 03:14
  • Ah yes, Dandelin spheres really are beautiful. Thanks for your help @Blue – Isaac Greene Jul 08 '18 at 11:39

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