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December 6th, 2009, 08:18 AM  #1 
Senior Member Joined: Sep 2008 Posts: 199 Thanks: 0  rectangular hyperbola
Prove that the chord joining the points P(cp , c/p) and Q (cq , c/q) on the rectangular hyperbola xy=c^2 has the equation pqy+x=c(p+q). I can prove this part. Given that the points P, Q and R lie on the hyperbola xy=c^2, prove that if PQ and PR are inclined equally to the coordinate axes, then QR passes through O. 
December 10th, 2009, 03:54 AM  #2 
Senior Member Joined: Sep 2008 Posts: 199 Thanks: 0  Re: rectangular hyperbola
can someone get me started . THanks .

December 10th, 2009, 02:26 PM  #3 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: rectangular hyperbola
The line PQ can be written and the line PR as so the slopes are 1/pq and 1/pr. If these are equal, then q=r, which makes Q the same point as R. So assume instead that one is the negative of the other. What is the consequence for the line QR ? 
December 10th, 2009, 02:45 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,633 Thanks: 2080 
Given distinct points P(cp, c/p), Q (cq, c/q), and R(cr, c/r) on the hyperbola, the slope of PQ is (c/q  c/p)/(cq  cp), i.e., 1/(pq), and similarly the slope of PR is 1/(pr). (These values are probably already known from the work done for the first part of the problem.) The above slopes are unequal, but may differ only in sign, in which case q = r. You already know QR has equation qry + x = c(q + r), so . . . (it's easy to finish from there). 
December 10th, 2009, 06:46 PM  #5  
Senior Member Joined: Sep 2008 Posts: 199 Thanks: 0  Re: Quote:
 

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