P4.3 - Coordinate geometry in the (x, y) plane

Parametric equations give both coordinates in terms of a third variable: $x=f(t)$ and $y=g(t)$. Each permitted value of $t$ produces a point $(x,y)$; changing $t$ traces the curve. A Cartesian equation removes $t$ and links $x$ directly to $y$.

To convert to Cartesian form, make $t$ or a simple function of $t$ the subject of one equation and substitute into the other. For trigonometric parameters, isolate $\sin t$ and $\cos t$ and use $\sin^2t+\cos^2t=1$. Finally translate the stated range of $t$ into restrictions on $x$ or $y$.

For $x=(t+3)/(t+1)$ and $y=2/(t+1)$ with $t>0$, notice that $x=1+2/(t+1)=1+y$. Hence $y=x-1$. Since $t+1>1$, $0<y<2$ and therefore $1<x<3$. The unrestricted line would contain points that the parameter never reaches.

For $x=3+2\cos t$ and $y=-1+4\sin t$, $$\cos t=\frac{x-3}{2},\qquad \sin t=\frac{y+1}{4}.$$ Therefore $$\frac{(x-3)^2}{4}+\frac{(y+1)^2}{16}=1.$$ The allowed interval for $t$ decides whether this represents the whole ellipse or only part of it.