Parametric form of the equation of normal to a parabola

Question

The equation of normal to a parabola is $y+ tx = 2at + at^3$ . This is a cubic equation in terms of $t$. That means we'll arrive at $3$ roots for $t$. But doesn't that mean that the normal will have $3$ intersection points with the parabola? How can a normal intersect a parabola thrice?

Answer 1

No, it means that up to three normals can be coincident. I suspect that you’re confusing what the variables in your equation represent. In particular, for any individual normal, $t$ is fixed and $x$ and $y$ vary—they are coordinates of points on the line. The equation on its own says nothing about how many intersections that line might have with the parabola.

On the other hand, if you view that equation as a cubic in $t$, then you are holding $x$ and $y$ constant—they are the coordinates of some fixed point on the plane, and not necessarily one that lies on the parabola. Since this cubic can have up to three distinct real roots, this means that there can be up to three normals to the parabola that pass through a given point on the plane.