Let $T$ be a planar curve that does not contain any circular arcs. Let $C$ be its osculating circle at the point $P$. Let $O$ be an intersection of $C$ with $T$ in a place other than $P,$ if it exists, and let it be $P$ itself if such a place doesn't exist. Let there be a constant $k$ and a line $L$ such that the directed distance between $O$ and $L$ is $k$ times the directed distance between $P$ and $L$ no matter where $P$ is chosen on the curve. What are the possible values of $k$ if we can choose $T?$
Making $T$ a parabola can give two possible values of $k.$ If $L$ is the line of symmetry, then $k = -3.$ If $L$ is the tangent at the vertex, then $k = 9.$
Here's an interactive osculating circle finder.