If the focus and directrix of a parabola are $(3, 5)$ and $x + y = 4$,then the coordinates of its vertex are

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$,respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
$(A)$ vertex is $\left(\frac{2a}{3}, 0\right)$
$(B)$ directrix is $x=0$
$(C)$ latus rectum is $\frac{2a}{3}$
$(D)$ focus is $(a, 0)$

The length of the latus rectum of a parabola whose focal chord $PSQ$ is such that $PS = 3$ and $QS = 2$ is

If $P$ is a point on the parabola $y^2=8x$ and $A$ is the point $(1,0)$,then the locus of the mid-point of the line segment $AP$ is

The two parabolas $y^2 = 4x$ and $x^2 = 4y$ intersect at a point $P$,whose abscissa is not zero,such that

The line $lx + my + n = 0$ will touch the parabola $y^2 = 4ax$,if