Let $P(2,4)$ and $Q(18,-12)$ be the points on the parabola $y^2=8x$. The equation of the straight line having slope $\frac{1}{2}$ and passing through the point of intersection of the tangents to the parabola drawn at the points $P$ and $Q$ is

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 $(a, 0)$

Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the feet of the perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola,then the area of the quadrilateral $PQMN$ is equal to:

Suppose the parabola $(y-k)^2 = 4a(x-h)$ has vertex $A$ and passes through $O = (0,0)$ and $L = (0,2)$. Let $D$ be an end point of the latus rectum. Let the $Y$-axis intersect the axis of the parabola at $P$. Then,$\angle PDA$ is equal to

If $0 \leq x \leq 5$,then the minimum distance from the point $(0, c)$ to the parabola $y = x^2$ is:

If the axes are rotated anticlockwise through an angle $90^{\circ}$,then the equation $x^2=4ay$ is changed to the equation