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If you really like geometry, why don't you try solving this interesting problem just for the fun of it?
Let point F and line d be the focus and directrix of a parabola. Choose any line through F which intersects the parabola in two points A and B. There is a line tangent to the parabola at A and another tangent line at B. Let C be the intersection of these two tangents.
Show (prove) each of the following statements:
1) C always lies on d (the directrix).
2) AC and BC (the two tangent lines) are always perpendicular.
3) FC and AB are always perpendicular.
I have already proven these a couple of ways and I'm not looking for detailed proofs - just let me know how you approached the problem and if you were successful. Did you use analytic geometry or not? Did you incidentally discover any other interesting facts while working on this? Do you know any other similar interesting problems?
Hint - Remember that rays which enter a parabola parallel to its axis reflect off the parabola toward its focus, and conversely, rays emitted from the focus reflect out parallel to the axis.
2 Answers
- 1 decade agoFavorite Answer
I used the following property:
1- Any point on parabola has similar distance to point F and D. Draw a line from A perpendicular to d and let H be the intersection of that line and d.
First I proved that the tangent at A bisects the angle between AF and AH.
Then I proved two triangles AFC and AHC are equal therefore angle CFA^ is 90 degress.
Then I draw a line from C to B and a line BK from B perpendicular to d (the intersection at K)
I proved Triangles BKC and BFH are equal and therefore BC bisects angle FBK, i.e., BC is tangent to parabola. The angle BFA can be proved 90 degree because sum of angles HAB and KBA are 180 and with the bisection properties sum of CAB and CBA becomes 90 degress. Then in triangle BCA the other angle becomes 180-90=90 degrees.
Other interesting propertied: C is in the middle of HK abd CH=CK=CF.
The problem can also be solved with analytical geometry needs some algebra
- 1 decade ago
Okay, so i have this all drawn out, and I don't see how it works, since if you have a at one point it would intersect d at a different point that if a was at another point, yet b could remain the same.
