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Can anyone see if this is right? (problem with discs and parabolas)

  • Thread starter jaumzaum
  • Start date
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My teacher proposed the following problem in the classroom as a challenge:
Consider 2 disks in the plane. Prove that you can always find 2 points exterior to the disks so that all the parabolas that pass through both points will intersect at least one of the disks.

I think I found a solution, can anyone check if it is right?
246465

246466
 

.Scott

Homework Helper
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I'm not sure about this.

What about this reasoning:
No two points on a parabola will have the same slope.
Therefore, picking two points - each on one of the spheres and each with the same slope will make it impossible for a parabola to be drawn that is tangent at both points. Thus any parabola passing through those two points will cross into one of the circles.

But, having picked those two points, moving the slightest distance away will both bring you into compliance with the rules - and allow a parabola to be drawn that avoids both circles.

It seems like a very marginal situation. You create the possibility of avoiding both disks as soon as you become "exterior" to the disk.
 

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