The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^2=4ax$ is a conic. The equation of the directrix of that conic is

Let $A_1, B_1, C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8x$ at $A_1$ and $B_1$,respectively. If $O=(0,0)$ and $C_1=(-4,0)$,then which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ The length of the line segment $OA_1$ is $4\sqrt{3}$
$(B)$ The length of the line segment $A_1 B_1$ is $16$
$(C)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(0,0)$
$(D)$ The orthocenter of the triangle $A_1 B_1 C_1$ is $(1,0)$

The length of the chord of the parabola $y^{2}=4ax$ $(a>0)$ which passes through the vertex and makes an acute angle $\alpha$ with the axis of the parabola is

$A$ point on the parabola $y^2 = 18x$ at which the ordinate increases at twice the rate of the abscissa is

Let the locus of the points from which the tangents drawn to $y = x^2$ make an angle of $45^{\circ}$ with each other be $16y^2 - 16x^2 + ky + 1 = 0$. Then $k$ is equal to:

Find the locus of the midpoint of the chord of the parabola $y^2 = 4x$ drawn from the vertex.