On determining the plane curves on which the length of the segment of the normal lines between the curve and the x axis are constant

Question

So, if $\alpha(t)$ is such a curve, then $g(r) = \alpha(t) + rn(t)$ is the normal line and $w(s) = \alpha(t) + s\alpha'(t)$ is the tangent line, where $n(t)$ is the orthogonal vector to $\alpha'(t)$. Let $P(x_0, y_0)$ be the point where the normal line intersects the $x$ axis and $Q = (x_0, y_0)$ be the vector from the origin to $P$, then I need to find the length of $\alpha(t) - Q$. I tried using coordinates, but as it usually happens, things got very messy. I would appreciate some help with finding an easier start point.

Answer 1

HINT: Assume the original curve $\alpha$ is arclength parametrized and consider the equation $$\big(\alpha(s)+\lambda(s)n(s)\big)\cdot (0,1) = 0.$$ What are you assuming about $\lambda(s)$? And what do you always do in differential geometry when you have an equation like this? :)

I'm not sure this is a very interesting question, but you should find two kinds of curves $\alpha$ for which this works.