Given an ellipse defined by its foci and major axis.
Prove that any line inside the ellipse not passing through the line segment joining the two foci when reflected off the ellipse would be tangent to the same inner ellipse as the initial line, with the same pair of foci.
Accordingly, if the line were to originally pass through the line segment joining the foci, the reflection on the ellipse would be tangent to the same hyperbola as the initial line, with the same pair of foci as the original ellipse.
The other possibility would be if the line started of from one focus, its reflection then passes through the other focus. (This degenerate case is known as the optical property of the ellipse, whose proof is much simpler.)
(In reply to Question.
by Vee-Liem Veefessional)
The order of posting of problems depends on several factors, one of which is the order of original submission onto the queue. Any given puzzle author/submitter is usually allocated only two slots out of ten in the queue in which Journeymen and Scholars can see them and vote on them. We try to extend down to the later-submitted problems, but those that were submitted earlier have some priority, especially if a given submitter, such as KS has offered so many puzzles for us.
As to why your later-submitted puzzle may have been posted before an earlier-submitted one, it may be that at the time that consideration was given, the earlier one did not yet have enough vote (thumbs up). We don't have access to back records of voting, so we can't look up to see if that's what happened. It could also be that, in an attempt to vary puzzle types from day to day, the later submitted one was seen to give greater variety at the time.
As you see there are various considerations.
Posted by Charlie
on 2010-05-06 11:15:49