Interval on parametric curves

Question

I have to plot the graph of $$\gamma(t)=(4\tan(t),9\sec(t))$$ for $t\in[0,\pi/2)\cup[\pi,3\pi/2)$.

I know the curve is an hyperbola, but I'd like to know if this interval of $t$ makes some difference in the graph itself.

Answer 1

Yes, the parametric curve $\gamma$ is a part of the hyperbola, $$\frac{y^2}{9^2}-\frac{x^2}{16^2}=1$$ For $t\in[0,\pi/2)$ we get the branch in the first quadrant and for $t\in[\pi,3\pi/2)$ we obtain the branch in the fourth quadrant. These two branches are symmetric with respect to the line $y=0$.