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wu :: forums    riddles    easy (Moderators: SMQ, ThudnBlunder, Grimbal, Icarus, towr, william wu, Eigenray)    Find the Limit « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Find the Limit  (Read 771 times) ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Find the Limit   TriangleLimit.gif « on: Jan 22nd, 2009, 7:14am » Quote Modify The triangle ABC below is isosceles with AB = BC. AX bisects angle CAB and X is the point of intersection of the angle bisector and the side CB. Let OC = 1 and let B move towards the side AC of the triangle along the perpendicular bisector BO.     1) What is the limiting length of OX as       a) -> 0?     b) -> /4?   2) What is the locus of X for 0 < < //4?   « Last Edit: Jan 22nd, 2009, 7:17am by ThudnBlunder » IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. SMQ wu::riddles Moderator Uberpuzzler     Gender: Posts: 2084 Re: Find the Limit   « Reply #1 on: Jan 22nd, 2009, 8:54am » Quote Modify Place O at the origin, and let X have coordinates (x, y).  From line AX we have y = (1 + x) tan , and from line CB we have y = (1 - x) tan 2.  Setting these equal and using the double angle formula for tan we find x = (1 + tan2 ) / (2 - tan2 ), y = 4 tan / (3 - tan2 ).  We can now evaluate 1 a) and 1 b) directly as 1/3 and 5 respectively.   --SMQ IP Logged --SMQ Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Find the Limit   « Reply #2 on: Jan 22nd, 2009, 12:36pm » Quote Modify ...which gives the hyperbola (3x+1)2 - 3y2 = 4. Interestingly, one of the foci is the point C.  The other is at (-5/3,0)   IP Logged Immanuel_Bonfils Junior Member     Posts: 114 Re: Find the Limit   « Reply #3 on: Jan 27th, 2009, 7:52am » Quote Modify No need of Analytic... From the bisector theorem we get  |CX| = (2 sec )/ (2 + sec ) where      = 2 .  Then the cosine law on OCX  triangle gives x2  = 1 + (4 sec 2 ) / (2 + sec )2  - 4 / ( 2 + sec ), where x =  |OX|.   Then comes traight the limits for  ->0 giving x ->1/3 and -> /2 giving x-> 5 IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------=> easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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