Recently, while using GeoGebra, I was able to generalize the idea of a parabolic vertex-kissing circuit I have achieved a wonderful result, but I cannot prove it. If anyone can prove it, please do so
Let $P$ be a parabola whose focus is $F$, whose directory is $L$, whose vertex is $O$, and whose axis of symmetry is $D$. The distance between $F$ and $L$ is equal to $d$.
Let $Q$ be an ellipse whose center is $S∈D$, one of its peaks $O$, and $M$ be one of the two vertices of $Q$ that does not lie on $D$, and let $k=OS/MS$.
Let $g=OS/d$
The largest possible ellipse $Q$ that has the value $k$ such that it touches $P$ at only one point is the ellipse that makes $g=k²$.
Any positive real number can be chosen as the value of the number $k$
The circle kissing the vertex of the parabola is a special case when $k=1$
Is this result known in advance? If so, please provide references that include this result
This is a linguistic description of what I have achieved: I have created the largest ellipse of all the ellipses that has a certain fixed ratio between its diagonals, which lies within the parabola, and the vertex of the parabola is one of its peaks, and the axis of symmetry of the parabola is one of its axes of symmetry, so that the parabola and the ellipse do not They have more than one point in common.
It is an ellipse whose ratio between the distance of its center and the vertex of the parabola and the distance between the focus and the index is equal to the square of the ratio between the two dimensions of the ellipse
This is a dynamic property observation drawing on GeoGebra. You can move $V$ to change the ratio $K$ and you can move $U$ to change the size of the ellipse while maintaining $K$. If you enlarge the ellipse enough then the points $H$ and $J$ will appear to intersect the ellipse with the parabola. We are interested in the case where These two points are inseparable.
You can monitor the values in the algebra section and see what happens. (This plot is formatted for $k>1$)
Best Answer
Your finding is not new: the limiting "kissing" ellipse is such that its curvature $\kappa$ at vertex $O$ is the same as the curvature of the parabola at $O$. In fact: $$ \kappa_\text{ellipse}={SO\over SM^2}, \quad \kappa_\text{parabola}={1\over d}, $$ where $d$ is the distance between focus and directrix of the parabola (note that $\kappa$ is in both cases the reciprocal of the latus rectum). Just multiply both curvatures by $SO$ to get your equality.