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#1 2014-03-08 14:45:56

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

hyperbolas and orthocenters DX

Let A, B, and C be three points on the curve xy = 1 (which is a hyperbola). Prove that the orthocenter of triangle ABC also lies on the curve xy = 1.

thanks!
hmm smile


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#2 2014-03-08 20:27:11

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: hyperbolas and orthocenters DX

hi thedarktiger,

This is how I'd do this:

Call the points (a,1/a)  (b,1/b)   and    (c,1/c)

The gradient from b to c is

Any line at right angles to (a line) will have gradient

so the equation of the altitude through a is:

Now I want the equation of another altitude to find where they cross.

It is not necessary to repeat the above.  It is ok just to say

similarly, the equation for the altitude through b is:     

If you want the practice, you could try getting this yourself.

Now, where do these lines cross.

It looked easiest to eliminate y.  I multiplied (i) by b, and equation (ii) by a.  Then subtraction eliminates the y terms.

Make x the subject.  Lots cancels leaving

It's interesting to note that this expression has 'equal' a, b and c bits in it.  This shows that the altitude through c wll also go through this point.

Now substitute that x back into (i) to get y.  You should get

If you now compute x times y you'll see you get 1.  So this point also lies on the hyperbola.

Bob

Last edited by Bob (2014-03-08 20:32:05)


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2014-03-11 00:08:13

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

Re: hyperbolas and orthocenters DX

Thank you!


Good. You can read.

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