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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 $y= \frac{-1}{pq}x + c\frac{p+q}{pq}$ and the line PR as $y= \frac{-1}{pr}x + c\frac{p+r}{pr}$ 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 $y= \frac{-1}{qr}x + c\frac{q+r}{qr}$ ?
 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
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 Originally Posted by skipjack 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).
thank you guys .

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### what is slope of ractangular hyperbola

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