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If i can prove that property,i won't need eccentricity for the equation.
Oh,but how did that site derive the first property?
has y = x as its tangent at infinity.
Please don't ask me to do so. I've just spent most of the day checking and filing paperwork and now my head is like this
Check this- http://www3.ul.ie/~rynnet/swconics/HP%27s.htm first property how was it derived???
I like the definition that asymptote is tangent to the curve at infinity.
I've got a better one, in that the curve gets closer and closer to y = x as x tends to infinity.
here's one to ponder:
Arhh. Now that is a question.
Bob,one last question ,what is the definition of an asymptote,there are many on the net,but which one is correct
I'm assuming you can use that in the same way we used the
for an ellipse.
From that you can show that the constant is 2a.
The focus and directrix properties will follow once you have the equation, but you might as well define some things in preparation.
eg. Define c to be ae for some e > 1
Start with (1) and substitute in what we have ( a, c etc)
Simplify as before.
Define b^2 = a^2(e^2 - 1) and the right equation should arrive after some work. (I haven't tried it yet )
This definition for b is actually equivalent to the formula you wanted to prove but the asymptote properties will come as a consequence rather than as a starting point.
I define a as the 1/2 of the distance of the two vertace and b as 1/2 of the line segment that is perpendicular to major axis,passes through center and its length corresponds to height of asymptotes over/under a vertex
That should enable you to introduce c.
Is that your starting point?
Then, how are you defining b ?
I think you want
e > 0
Sorry for not being clear,c is distance of foci from center.if i assume a^2+b^2=c^2 i can derive the equation but how do i prove a^2+b^2=c^2?