The distance between the vertex and the focus of the parabola $x^2-2x+3y-2=0$ is

The coordinates of the focus of the parabola described parametrically by $x=5t^2+2, y=10t+4$ (where $t$ is a parameter) are

The equation of the tangent to the parabola $y = x^2 - x$ at the point where $x = 1$ is:

The equation of the normal to the parabola $x^2 = 8y$ at $x = 4$ is

What is the locus of the vertices $(x, y)$ of the parabolas $y = \frac{a^3x^2}{3} + \frac{a^2x}{2} - 2a$?

Let $A$ be the focus of the parabola $y^{2}=8x$. Let the line $y=mx+c$ intersect the parabola at two distinct points $B$ and $C$. If the centroid of the triangle $ABC$ is $(\frac{7}{3},\frac{4}{3})$,then $(BC)^{2}$ is equal to: