Parabolas which are osculating to a given one

Question

I was trying to do this problem in $A(\mathbb{R}^2)$which consists in finding the parabolas which are osculating to $1 +2x +2y +x^2 = 0$ in the line at infinity and contain the point (1,-1). I only got some examples of parabolas which have these properties but not a general expression of them.

Answer 1

So here's a sketch of how you might proceed (with details left for you to work out).

(1) If we're looking for parabolas that meet the line at infinity at the point $[0,1,0]$, they will all have the form $$y=ax^2+bx+c \quad \text{for some scalars } a,b,c. \tag{$\star$}$$

(2) The condition that the parabola passes through $(1,-1)$ gives you one linear equation on $a,b,c$.

(3) To deal with the question of second order contact at infinity, you want to homogenize and then look in the coordinates where $y=1$, near the origin in $xz$-coordinates. The given parabola has the equation $2z=(x+z)^2$, and, differentiating implicitly, we find $z'=0$ and $z''=-1$ at the origin. Now do similarly for our equation ($\star$) and setting $z''=-1$ will give you another linear equation in $a,b,c$. ($z'=0$ is automatic because all our parabolas are tangent to the line at infinity.)

At any rate, you will end up with a one-parameter family of solutions.