TITLE: closest point to on $y=1/x$ to a given point
QUESTION [7 upvotes]: I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula but it leads to an order-4 polynomial...

REPLY [2 votes]: I tried the direct approach of pluggin in y=1/x into the distance formula but it leads to an order-4 polynomial...

That is not a mistake. The ellipse has two maxima and two minima of the distance to $P$ for most locations of point $P$.  This leads to a degree $4$ polynomial when solving for the coordinates of the extrema.  There is no special geometric relationship between the distance extrema that might make the polynomial easier to solve than the general quartic equation.