A Question on Ellipses, Part 2

In the last post, I posed the question:

What is the set of points of intersection of a normal line of an ellipse with its major axis, barring the normal lines at the endpoints of the major axis?

The most common responses I’ve gotten have been: (1) the entire major axis, or (2) the segment connecting the foci. These are reasonable answers, since it must be a subset of the major axis, and what other important subsets of the major axis does an ellipse have? Let’s see if either of these is right.

The equation of an ellipse takes the form $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$ (at least if its centered at the origin, and why not?). It will be convenient if the major axis is horizontal, so we will assume that $a>b>0$.

Differentiating both sides of the ellipse implicitly gives $$\dfrac{2x}{a^2}+\dfrac{2yy’}{b^2}=0,$$ so $y’=-\dfrac{b^2x}{a^2y}$. At any point $(x_0,y_0)$ on the ellipse, the slope of the tangent line is $$m_{\text{tangent}}=-\dfrac{b^2x_0}{a^2y_0}.$$ Taking the opposite reciprocal, we get that the slope of the normal line at $(x_0,y_0)$ is $m_{\text{normal}}=\dfrac{a^2y_0}{b^2x_0},$ and so the equation of the normal line is $$y=\dfrac{a^2y_0}{b^2x_0}(x-x_0)+y_0.$$ This line intersects the major axis at $\dfrac{a^2-b^2}{a^2}x_0$, and since the points on the ellipse satisfy $-a<x_0<a$, we know that the points of intersection of the normal lines and the major axis are on the segment from $\left(-\dfrac{a^2-b^2}{a},0\right)$ and $\left(\dfrac{a^2-b^2}{a},0\right)$.

The foci of the ellipse are at $\left(-\sqrt{a^2-b^2},0\right)$ and $\left(\sqrt{a^2-b^2},0\right)$, and so the segment of intersections does not quite reach from focus-to-focus.

Why not?