1. ## Rectangular Hyperbola

Can someone please explain to me why the parametric coordinates $\displaystyle \ (a\cosh t,a\sinh t) \$ can only represent one branch of the rectangular hyperbola with the equation $\displaystyle \ 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 $\displaystyle \ P(a\cosh t,a\sinh t) \$, where t>0, passes through the point $\displaystyle \ (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?

2. Can someone please explain to me why the parametric coordinates $\displaystyle \ (a\cosh t,a\sinh t) \$ can only represent one branch of the rectangular hyperbola with the equation $\displaystyle \ 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

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

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

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

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

This curve is called a hyperbola and therefore sinh and cosh are called hyperbolic functions.

3. 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?