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January 5th, 2012, 04:47 PM  #1 
Newbie Joined: Jan 2012 Posts: 28 Thanks: 0  Find the number of chords
A chord of length 16 is perpendicular to a diameter at point A. If the circle's diameter is 30, find the number of chords of integral length (other than the diameter) that contain A.

January 5th, 2012, 08:49 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,208 Thanks: 516 Math Focus: Calculus/ODEs  Re: Find the number of chords
Using coordinate geometry, where the center of the circle is at the origin, the circle is given by: Taking point A as (A,0) where 0 < A we set: The family of lines passing through A is: where to exclude the diameter. So we then substitute for y into the circle's equation: So we then require: where and 2n is the chord length. In order for m to be real and nonzero, we require: We find and for each n there are two chords for the two roots of m, and adding the given vertical chord makes a total of: 2(30  17) + 1 = 27 chords passing through A of integral length, excluding the diameter, represented by n = 15 and m = 0. In retrospect, the answer seems so obvious now. 
January 6th, 2012, 06:39 PM  #3 
Newbie Joined: Jan 2012 Posts: 28 Thanks: 0  Re: Find the number of chords
Thanks for your try but the answer is only between 26 and 1. I hope you can explain it to me. Thanks

January 6th, 2012, 07:15 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,208 Thanks: 516 Math Focus: Calculus/ODEs  Re: Find the number of chords
Perhaps they are also excluding the given chord as well as the diameter? edit: I notice on another forum the length of the vertical chord is given as 18...which is it, 16 or 18? 
January 6th, 2012, 09:29 PM  #5 
Newbie Joined: Jan 2012 Posts: 28 Thanks: 0  Re: Find the number of chords
It was my mistake that it is 18 . Thanks I didn't notice that too.

January 6th, 2012, 09:30 PM  #6 
Newbie Joined: Jan 2012 Posts: 28 Thanks: 0  Re: Find the number of chords
Can you show how your solve this again with 18 instead of 16. Thanks

January 6th, 2012, 09:42 PM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,208 Thanks: 516 Math Focus: Calculus/ODEs  Re: Find the number of chords
If we exclude the given chord (minimum length) and the diameter (maximum length) then we have 2 chords of length {19, 20,...28,29} = 22 chords.

January 7th, 2012, 08:26 AM  #8 
Newbie Joined: Jan 2012 Posts: 28 Thanks: 0  Re: Find the number of chords
The number of chords of integral length (other than the diameter) that contain A is 22??


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