The line $y = mx + c$ touches the parabola $x^2 = 4ay$ if:

For the parabola represented in the parametric form by $x=t^2+t+1$ and $y=t^2-t+1$,the length of the latus rectum is

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^{2}=4x$ be the curve $S$. Let $P$ be any point on $S$. Then the locus of the point,which internally divides $OP$ in the ratio $3:1$,is:

If $(a, b)$ is the midpoint of a chord passing through the vertex of the parabola $y^2 = 4x$,then:

Find the equation of the directrix of the parabola $y^2 = 12x$.

An equilateral triangle is inscribed in the parabola $y^2 = 8x$,with one of its vertices at the vertex of the parabola. Then,the length of the side of that triangle is