On a recent ACT practice problem that I did, I came across this problem:
Given the parametric equations $x=\sec\theta$ and $y=\tan\theta$, which of the following is the graph of the points $(x,y)$?
(A)$\quad$circle
(B)$\quad$ellipse
(C)$\quad$parabola
(D)$\quad$hyperbola
(E)$\quad$None of these
I have plotted this on Desmos (an online graphing tool) and determined that the answer is (D), but I am not too sure how to arrive at this answer. Note that this is a problem that was found on ACT, a test meant to measure high school students' readiness for college, meaning that the solution shouldn't be too complicated, and it shouldn't take too long either. How can I solve this problem in a reasonable amount of time(say, 60 to 90 seconds), and if you want to, could you also provide some tips for problems similar to this problem?
I don't know if this is a feasible way or not, but my thought process while doing this problem was to plug in arbitrary values for $\theta$ then plotting the resulting points to obtain the answer.
To fill out the details of Matthew's suggestion: note that, since $x = \sec \theta$ and $y = \tan \theta$, then perhaps some identity relating secant and tangent is worth exploring. We have the identity
$$\sec^2 \theta = \tan^2 \theta + 1$$
In terms of $x,y$, that identity is
$$x^2 = y^2 + 1$$
That is,
$$x^2 - y^2 = 1$$
This describes a hyperbola, centered at the origin, with transverse/conjugate axis lengths of $1/2$, which opens towards left/right half-planes.
(Though in the case of a multiple-choice question like this, you could just note "oh hey I have $x^2 - y^2 = 1$ and thus it's a hyperbola" and that will be sufficient.)