# Hyperbolas: Chicken-and-egg problem with c^2 = a^2 + b^2 Classic List Threaded 4 messages Open this post in threaded view
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## Hyperbolas: Chicken-and-egg problem with c^2 = a^2 + b^2

 Hello Carl, I refer to pages 533-534 of Precalculus where you (you and Jeff collectively) derive the equation x^2/a^2 – y^2/b^2 = 1 for the simplest case of a horizontal hyperbola centred at (0, 0). There is one sticking point for me in the derivation, namely, where you set c^2 = a^2 + b^2.  You prove the analogous equation for horizontal ellipses (a^2 = b^2 + c^2) before relying on it to derive the basic equation of x^2/a^2 + y^2/b^2 = 1 for the latter.  However, the justification for c^2 = a^2 + b^2 for hyperbolas is a mystery to me. I am aware that there is a natural reflective property relating to the foci of a hyperbola, so the relationship between a, b and c is something naturally occurring and there to be discovered – not something that is set by fiat. I've searched the Internet trying to find an answer, and all I've found is that there are other people like me who've done the same thing to no avail.  It's as if nobody wants to talk about the answer, whatever it is.  One site acknowledged the issue with a comment that it's all too hard, and just accept that c^2 = a^2 + b^2 for hyperbolas and move on. Every derivation of x^2/a^2 – y^2/b^2 = 1 that I've seen relies on the (to me) unproven proposition that c^2 = a^2 + b^2. I found a proof for c^2 = a^2 + b^2 on YouTube, but right at its end it relied on x^2/a^2 – y^2/b^2 = 1 as a prerequisite fact.  I also devised my own rather simpler proof for c^2 = a^2 + b^2, but then realised that I was relying on the truth of y = +–b/a sqrt(x^2 – a^2), which of course is derived from x^2/a^2 – y^2/b^2 = 1. So, all I've seen so far is a chicken-and-egg relationship between the two derivations.  You need to accept the truth of c^2 = a^2 + b^2 in order to derive x^2/a^2 – y^2/b^2 = 1, but you also need to accept the truth of x^2/a^2 – y^2/b^2 = 1 in order to prove c^2 = a^2 + b^2. There were some sticking points for me when I was working through the chapter on polynomials, but I was eventually able to resolve all of them to my satisfaction so that I felt "at peace" with the propositions in question, and could move on.  However, I'm not seeing any light at the end of the tunnel with this particular issue with hyperbolas. Carl, can you shed some light on this, or are we dealing with an unspeakably awful Sasquatchesque monster here?
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## Re: Hyperbolas: Chicken-and-egg problem with c^2 = a^2 + b^2

 Administrator Hello! Great question.   Basically our derivation writes the equation of the hyperbola using the distance definition in terms of the parameters "a" and "c".  (Our diagram uses the letter "b" in the label, but you can ignore that picture for now.  It doesn't figure into the derivation in terms of "a" and "c".) We then use an end-behavior argument to show the ratio of y/x in the hyperbola is (+/-) sqrt(c^2-a^2) /a. This means the hyperbola is asymptotic to a rectangle whose diagonals have ratio  (+/-) sqrt(c^2-a^2)/a. The key step is here:  we  *define* b =  sqrt(c^2-a^2), since the slope of the diagonals is (+/-) sqrt(c^2-a^2) /a = (+/-) b/a, this means the  endpoints of the conjugate axis are (0, (+/-)b) . So... to recap:  we use the definition of hyperbola to get the equation in terms of "a" and "c".  We then note the hyperbola is asymptotic to the lines y =[ (+/-) sqrt(c^2-a^2) /a] x so we call the numerator of the slope, sqrt(c^2-a^2) = b. Does that help?