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

What is the common tangent to the parabola $y^{2} = 8ax$ and the circle $x^{2} + y^{2} = 2a^{2}$?

Let $O$ be the vertex of the parabola $y^2 = 4x$ and its chords $OP$ and $OQ$ are perpendicular to each other. If the locus of the mid-point of the line segment $PQ$ is a conic $C$, then the length of its latus rectum is:

The tangents drawn at the endpoints of the latus rectum of a parabola $S = 0$ intersect on the line $x + y = 2$. If $(3, 2)$ is the focus of the parabola,then the axis of the parabola $S = 0$ is:

The focal chord to $y^2 = 16x$ is tangent to $(x - 6)^2 + y^2 = 2$. Then,the possible values of the slope of this chord 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 $(a, 0)$