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Synthetical Solution hidden behind Analytic Geom. notations
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freemind
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#1
Synthetical Solution hidden behind Analytic Geom. notations
Moldova National MO 2008, 12 Grade, Problem 3
In the usual coordinate system xOy a line d intersect the circles C_1: (x+1)^2+y^2=1 and C_2: (x-2)^2+y^2=4 in the points A,B,C and D (in this order). It is known that A\left(-\frac32,\frac{\sqrt3}2\right) and \angle{BOC}=60^{\circ}. All the Oy coordinates of these 4 points are positive. Find the slope of d.
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PostPosted: Sat Mar 01, 2008 4:41 pm
Nick Rapanos
Yang-Mills Theory
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Re: Synthetical Solution hidden behind Analytic Geom. notations
Moldova National MO 2008, 12 Grade, Problem 3
freemind wrote:
In the usual coordinate system xOy a line d intersect the circles C_1: (x + 1)^2 + y^2 = 1 and C_2: (x - 2)^2 + y^2 = 4 in the points A,B,C and D (in this order). It is known that A\left( - \frac32,\frac {\sqrt3}2\right) and \angle{BOC} = 60^{\circ}. All the Oy coordinates of these 4 points are positive. Find the slope of d.


Hello Andrew, here is my solution.

Firstly one can show easilly that \angle{KLA} = 60^o and therefore \angle{OBC} = 60^o\Longrightarrow \angle{OCB} = 60^o and finally \angle{OFD} = 60^o. Now the triangle GDF is isosceles and \angle{GFD} = 60^o, which means that the triangle GFD is equilateral. Therefore the abscissa of point D must be equal to OM where M the midpoint of the segment GF, so D = (3,\sqrt {3}) and now d = \frac {DT}{AT} = \frac {\sqrt {3}}{9}.

I hope that my calculations are ok, but forgive me if not.

Nick Rapanos
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PostPosted: Sat Mar 01, 2008 6:35 pm
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