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

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.

$$\vec{r} = \vec{a} + \lambda\vec{b} + \lambda^2\vec{c}$$ 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.