The point at which the line $y = mx + c$ touches the parabola $y^2 = 4ax$ is

The equations of two sides of a variable triangle are $x = 0$ and $y = 3$,and its third side is a tangent to the parabola $y^2 = 6x$. The locus of its circumcentre is:

Let $E$ denote the parabola $y^2=8x$. Let $P=(-2,4)$,and let $Q$ and $Q^{\prime}$ be two distinct points on $E$ such that the lines $PQ$ and $PQ^{\prime}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) $TRUE$?
$(A)$ The triangle $PFQ$ is a right-angled triangle
$(B)$ The triangle $QPQ^{\prime}$ is a right-angled triangle
$(C)$ The distance between $P$ and $F$ is $5\sqrt{2}$
$(D)$ $F$ lies on the line joining $Q$ and $Q^{\prime}$

Which of the following points lie on the parabola $x^2 = 4ay$?

If two normals are drawn from a point to the parabola $y^{2} = 4ax$ such that they make angles $\alpha$ and $\beta$ with the axis,and $\tan \alpha \cdot \tan \beta = 2$,find the locus of the point.

Let the tangent and normal at any point $P(at^2, 2at)$, $(a > 0)$, on the parabola $y^2 = 4ax$ meet the axis of the parabola at $T$ and $G$ respectively. Then the radius of the circle through $P, T$ and $G$ is