What's the standard form of the equation of a line of a slanted parabola?

Question

I have been trying to figure out the general form of a slanted parabola, but what I've gotten doesn't look like it would be accurate:$$(x-h)^2+(y-k)^2=\dfrac{d}{\sqrt{h}}$$Where $(h,k)$ is the focus, and $d$ is the directrix. Apparently, slanted conics have a mixed term of the form $Kxy$, but I cannot get that to apply.

Answer 1

I may have an answer, "Slanted parabolas are also parabolas". Sometime ago, I was trying to find a vector equation for parabola,

$$\vec{r} = \vec{a} + \lambda\vec{b} + \lambda^2\vec{c}$$ you might notice that, if $\vec{b}$ and $\vec{c}$ are not perpendicular, then you will get a slanted parabola.

e.g. let $\vec{a} = \vec{0}$, and $\vec{b}=\widehat{i}$ and $\vec{c}=c\widehat{i}+\widehat{j}$

For past few days, I've been playing with them, and I proved that, slanted parabolas are also parabolas which are rotated, and stretched by some amount. and I've proved that, for any vector $\vec{b}$, and $\vec{c}$ the above vector equation will always yield a parabola. and if you want some standard equation for slanted parabola, I can give you this,

$$x(\lambda) = \lambda + c\lambda^2$$ $$y(\lambda) = \lambda^2$$ draw the x-y graph for any fixed value of "c", you'll get a slanted parabola (which I proved is a parabola, you can try that yourself, just rotate the vector b and c such that c points upwards), you can scale the slanted parabolas to get all the possible slanted parabolas.

If you want a python matplotlib program which animates this, comment, I'll post it on github. And if you want the proof, I was talking about, I haven't written it in proper mathematical terms, but I can try to post that as well if you want.