The intercept of the normal to an ellipse on its major axis, as its intersection point on the ellipse goes closer and closer to the vertex, reaches a maximum of $\pm\frac{a^2-b^2}{a}$ for a standard ellipse (at the vertex). This can be arrived at by using limits, and is illustrated here graphically (the farthest intercept points are in green, the foci in black) ;
This wasn't intuitively obvious to me at first; I had the (in hindsight, moronic) notion that the intercept should have gone to infinity as the slope approached 0.
I'd like to know if there's a term for this; a moving line (here, moving because it's parameterized) pivoting around a point on another line, and if there are any related geometric results (not concerning just ellipses, but as general results), since it wasn't intuitively apparent to me at first, and I only noticed it while graphing it out.
You materialize in this way the envelope of normals, called the evolute of the initial curve. The evolute of a curve can be described in an equivalent way as the locus of centers of curvature of this curve.
The evolute of the ellipse is known to be a "compressed" astroid with extreme points looking as spikes, 2 of them on the main axis, the 2 others on the secondary axis of the ellipse (see here).
The reciprocal concept is called the involute: the ellipse is the involute of the compressed astroid above.