If $5x - 2y + k = 0$ is a tangent to the parabola $y^2 = 6x$,then their point of contact 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:

$A$ beam is supported at its ends by supports which are $12 \, m$ apart. Since the load is concentrated at its centre,there is a deflection of $3 \, cm$ at the centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection $1 \, cm$?

The focus and directrix of the parabola ${x^2} = - 8ay$ are

Let $P$ be a point on the parabola $y^2 = 4ax$,where $a > 0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$,where $F$ is the focus of the parabola,is $120$. If the slope $m$ of the normal and $a$ are both positive integers,then the pair $(a, m)$ is

The normal to the curve $x^{2}=4y$ passing through $(1,2)$ is