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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!

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

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)

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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: hyperbolas and orthocenters DX

Thank you!

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Good. You can read.