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Math Help - Rectangular Hyperbola

  1. #1
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    Rectangular Hyperbola

    Can someone please explain to me why the parametric coordinates \<br />
(a\cosh t,a\sinh t)<br />
\ can only represent one branch of the rectangular hyperbola with the equation \<br />
x^2  + y^2  = a^2 <br />
\, and what would the parametric coordinates of the other branch be?

    I'm stuck on a question relating to the rectangular hyperbola, which is:
    The normal at \<br />
P(a\cosh t,a\sinh t)<br />
\, where t>0, passes through the point \<br />
(4a,0)<br />
\. Find the non-zero value of t, giving your answer as a natural logarithm.

    I must admit I have absolutely no idea how to even begin the question. Can someone please enlighten me?
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  2. #2
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    Can someone please explain to me why the parametric coordinates \<br />
(a\cosh t,a\sinh t)<br />
\ can only represent one branch of the rectangular hyperbola with the equation \<br />
x^2  + y^2  = a^2 <br />
\, and what would the parametric coordinates of the other branch be?
    If t is any real number, then the point (cos(t),sin(t)) lies on the circle

    x^{2}+y^{2}=1 because cos^{2}(t)+sin^{2}(t)=1.

    For this reason sine and cosine are circular functions. So, in the same light,

    for any real number t the point (cosh(t),sinh(t)) lies on the curve

    x^{2}-y^{2}=1 because cosh^{2}(t)-sinh^{2}(t)=1.

    The other side is just (-cosh(t), -sinh(t))

    This curve is called a hyperbola and therefore sinh and cosh are called hyperbolic functions.
    Last edited by galactus; November 24th 2008 at 06:38 AM.
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  3. #3
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    The first part of the question makes a lot more sense now, but I still don't know how to do the seond part. Could someone explain to me please?
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