= Solution
Write $z=X+iY$. Its coordinates are $X=t$, $Y=1/t$, so $XY=1$. For the given positive range of $t$, this traces a segment of the first-quadrant <hyperbola>, moving rightward and downward:
$$
\boxed{z:\ (1/2,2)\longrightarrow(2,1/2).}
$$
The <complex conjugate> is $z^*=t-i/t$. It is the reflection in the <real axis>, tracing $XY=-1$ in the fourth quadrant, rightward and upward:
$$
\boxed{z^*:\ (1/2,-2)\longrightarrow(2,-1/2).}
$$
Neither trajectory crosses either axis, because $t$ and $1/t$ remain nonzero. The left panel of the figure marks the endpoints and directions.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1-q11-complex-trajectories.png]
{title=Trajectories under <complex conjugation>, squaring and fourth powers, with endpoints and axis crossings}
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