If the point $(a, 2a)$ is an interior point of the region bounded by the parabola $y^2 = 16x$ and the double ordinate through the focus,then

If the locus of a point which divides a chord with slope $2$ of the parabola $y^2=4x$ internally in the ratio $1:3$ is a parabola,then its vertex is

Find the length of the chord of the parabola $x^2 = 4ay$ which passes through the vertex and has a slope of $\tan \alpha$.

If the line $y = 4 + kx$,$k > 0$,is the tangent to the parabola $y = x - x^{2}$ at the point $P$ and $V$ is the vertex of the parabola,then the slope of the line through $P$ and $V$ is

Let the tangent to the parabola $S: y^{2}=2x$ at the point $P(2,2)$ meet the $x$-axis at $Q$ and the normal at $P$ meet the parabola $S$ at the point $R$. Then the area (in $sq. \ units$) of the triangle $PQR$ is equal to:

The equation of the tangent to the parabola $y^2=8x$,which is parallel to the line $4x-y+3=0$,is