How to convert parametric equations to Cartesian form?

Question

I've been working on converting parametric equations into Cartesian form, but can't figure this out.

$$x = \frac{t^2+1}{t^2-1}$$ $$y = \frac{2t}{t^2-1}$$

How do I covert that to Cartesian? Any help would be most appreciated.

Answer 1

Since $(t^2 - 1)x = t^2 + 1$ and $(t^2 - 1)y = 2t$, $$(t^2 - 1)^2 x^2 - (t^2 - 1)^2 y^2 = (t^2 + 1)^2 - 4t^2 = (t^4 + 2t^2 + 1) - 4t^2 = t^4 - 2t^2 + 1 = (t^2 - 1)^2.$$

Dividing through by $(t^2 - 1)^2$, we obtain $$x^2 - y^2 = 1$$

Answer 2

In general, you need to eliminate $t$ with the 2 equations.

For this one, we get a shortcut.

$$x+y={t+1 \over t-1}$$ $$x-y={t-1 \over t+1}$$

Then, obviously,

$$(x+y)(x-y)=1$$ $$(x^2-y^2)=1$$