= Solution
The <curve> has $x\ne0$ and can be written $y=1/x-x$. It is a <hyperbola> with vertical <asymptote> $x=0$ and oblique <asymptote> $y=-x$, crossing the horizontal axis at $(\pm1,0)$. It is symmetric under $(x,y)\mapsto(-x,-y)$.
On each branch $dy/dx=-1/x^2-1<0$. For $x>0$, the branch descends from $+\infty$ as $x\downarrow0$, passes through $(1,0)$, and approaches $y=-x$ from above as $x\to+\infty$. The negative branch follows by the central symmetry.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1-hyperbola-nearest-point.png]
{title=The two hyperbola branches and their asymptotes, with the nearest point on the positive-horizontal branch marked}
Back to article page
