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 January 5th, 2012, 03:47 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, 07:49 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 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: $x^2+y^2=15^2$ Taking point A as (A,0) where 0 < A we set: $A^2+8^2=15^2$ $A^2=23\cdot7=161$ $A=\sqrt{161}$ The family of lines passing through A is: $y=m(x-A)=m$$x-\sqrt{161}$$$ where $m\ne0$ to exclude the diameter. So we then substitute for y into the circle's equation: $x^2+$$m\(x-\sqrt{161}$$\)^2=15^2$ $x^2+m^2$$x^2-2\sqrt{161}x+161$$=15^2$ $$$m^2+1$$x^2-2m^2\sqrt{161}x+161m^2-15^2=0$ $x=\frac{2m^2\sqrt{161}\pm\sqrt{4m^4\cdot161+4$$m^2 +1$$$$15^2-161m^2$$}}{2$$m^2+1$$}$ $x=\frac{m^2\sqrt{161}\pm\sqrt{161m^4+$$m^2+1$$$$15 ^2-161m^2$$}}{m^2+1}$ $x=\frac{m^2\sqrt{161}\pm\sqrt{8^2m^2+15^2}}{m^2+1}$ $y=m$$\frac{-\sqrt{161}\pm\sqrt{8^2m^2+15^2}}{m^2+1}$$$ So we then require: $\sqrt{$$\frac{m^2\sqrt{161}+\sqrt{8^2m^2+15^2}}{m^ 2+1}-\frac{m^2\sqrt{161}-\sqrt{8^2m^2+15^2}}{m^2+1}$$^2+$$m\(\frac{-\sqrt{161}+\sqrt{8^2m^2+15^2}}{m^2+1}$$-m$$\frac{-\sqrt{161}-\sqrt{8^2m^2+15^2}}{m^2+1}$$\)^2}=2n$ where $n\in\mathbb N$ and 2n is the chord length. $\sqrt{$$\frac{2\sqrt{8^2m^2+15^2}}{m^2+1}$$^2+m^2\ (\frac{2\sqrt{8^2m^2+15^2}}{m^2+1}\)^2}=2n$ $\sqrt{\frac{8^2m^2+15^2}{m^2+1}}=n$ $\frac{8^2m^2+15^2}{m^2+1}=n^2$ $m^2=\frac{n^2-15^2}{8^2-n^2}$ In order for m to be real and non-zero, we require: $0>\frac{n^2-15^2}{n^2-8^2}$ We find $17\le 2n\le29$ 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, 05:39 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, 06:15 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 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, 08:29 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, 08:30 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, 08:42 PM #7 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 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, 07:26 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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