What is the equation of the normal to the parabola $y^2 = 4ax$ at the point $(\frac{a}{m^2}, \frac{2a}{m})$?

If the line $y=x$ is a tangent to the parabola $y=ax^{2}+bx+c$ at the point $(1,1)$ and the curve passes through $(-1,0)$,then

What are the parametric equations of the parabola $y^2 = 8x$?

Let $O$ be the vertex of the parabola $x^{2}=4y$ and $Q$ be any point on it. Let the locus of the point $P$,which divides the line segment $OQ$ internally in the ratio $2:3$,be the conic $C$. Then the equation of the chord of $C$,which is bisected at the point $(1, 2)$,is:

For any non-zero real value of $m$,the equation of the parabola to which the line $m x-y+10+m^2=0$ is a tangent,is

What is the focus of the parabola $y^2 - x - 2y + 2 = 0$?