wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> putnam exam (pure math) >> Reflective Circle
        
(Message started by: THUDandBLUNDER on Dec 14th, 2004, 7:16am)
Title: Reflective Circle
Post by THUDandBLUNDER on Dec 14th, 2004, 7:16am
I think this ought to be in Hard, but hey, it's too mathy.

Given a circle C and two points A and B external to it, find the point(s) on the circle such that both AC and BC make equal angles with the tangent to the circle passing through point C.
Title: Re: Reflective Circle
Post by Aryabhatta on Dec 14th, 2004, 9:03am

on 12/14/04 at 07:16:44, THUDandBLUNDER wrote:
I think this ought to be in Hard, but hey, it's too mathy.

Given a circle C and two points A and B external to it, find the point(s) on the circle such that both AC and BC make equal angles with the tangent to the circle passing through point C.


Did you mean: Given a circle and two points A and B, find all points C on the circle such that AC and BC make equal angles with tangent to the circle at C? (You have named the circle C which makes it confusing to read).

One other question, do A and B have to be on the same side of the tangent? Also the angle AC makes with the tangent seems not clearly defined.. which one do we take? x or 180-x? My guess (based on the title of this thread) is you want the tangent to be the segement PCQ, with C in between P and Q and the corresponding angles are ACP and BCQ.
Title: Re: Reflective Circle
Post by JocK on Dec 14th, 2004, 12:05pm
If M is the centre of the circle, then the reflection point C on the circle satisfies: [angle] MCA =  [angle] BCM.

You want this equation parametrised in some way? (Don't see the problem yet...) And yes, there is always two solutions, at most one corresponds to an "outside reflection".
Title: Re: Reflective Circle
Post by THUDandBLUNDER on Dec 14th, 2004, 7:35pm

on 12/14/04 at 09:03:17, Aryabhatta wrote:
Did you mean: Given a circle and two points A and B, find all points C on the circle such that AC and BC make equal angles with tangent to the circle at C? (You have named the circle C which makes it confusing to read).

One other question, do A and B have to be on the same side of the tangent? Also the angle AC makes with the tangent seems not clearly defined.. which one do we take? x or 180-x? My guess (based on the title of this thread) is you want the tangent to be the segement PCQ, with C in between P and Q and the corresponding angles are ACP and BCQ.

Sorry about the confusing double use of 'C'.

Physically, the problem asks for a method to find the point(s) on the boundary of a circular pool table at which the cue ball must be aimed, such that it hits the target ball after one bounce off the cushion. (For example, can it be constructed using only ruler and compasses?) To put it yet another way, find an isosceles triangle whose sides pass through two given points inside a given circle.


Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board