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January 5th, 2012, 03:47 PM   #1
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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.
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January 5th, 2012, 07:49 PM   #2
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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 non-zero, 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.
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January 6th, 2012, 05:39 PM   #3
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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
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January 6th, 2012, 06:15 PM   #4
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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?
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January 6th, 2012, 08:29 PM   #5
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Re: Find the number of chords

It was my mistake that it is 18 . Thanks I didn't notice that too.
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January 6th, 2012, 08:30 PM   #6
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Re: Find the number of chords

Can you show how your solve this again with 18 instead of 16. Thanks
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January 6th, 2012, 08:42 PM   #7
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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.
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January 7th, 2012, 07:26 AM   #8
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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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