Find the locus of the point of intersection of perpendicular tangents to the parabola $y^2 - 6y + 24x - 63 = 0$.

Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $x^{2}=6y$.

Find the equation of the normal to the parabola $y^2 + 12x = 0$ at the upper end of its latus rectum.

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 $(a, 0)$

From a point $(d, 0)$,three normals are drawn to the parabola $y^{2} = x$. Then:

Find the point where the normal drawn from the upper end of the latus rectum of the parabola $y^2 = -12x$ intersects the axis.