# Rectangular Hyperbola

• Feb 12th 2008, 10:27 AM
free_to_fly
Rectangular Hyperbola
Can someone please explain to me why the parametric coordinates $\
(a\cosh t,a\sinh t)
\$
can only represent one branch of the rectangular hyperbola with the equation $\
x^2 + y^2 = a^2
\$
, 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 $\
P(a\cosh t,a\sinh t)
\$
, where t>0, passes through the point $\
(4a,0)
\$
. 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?
• Feb 12th 2008, 11:41 AM
galactus
Quote:

Can someone please explain to me why the parametric coordinates $\
(a\cosh t,a\sinh t)
\$
can only represent one branch of the rectangular hyperbola with the equation $\
x^2 + y^2 = a^2
\$
, 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.
• Feb 12th 2008, 01:54 PM
free_to_fly
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?