If the segment of the parabola $y^2 = 4ax$ intercepted by its directrix subtends an angle $\theta$ at its focus,then $\theta = \dots$

If all the vertices of an equilateral triangle lie on the parabola $y^2=16x$ and one of them coincides with the vertex of that parabola,then the length of the side of that triangle is

The coordinates of the ends of a focal chord of a parabola $y^2 = 4ax$ are $(x_1, y_1)$ and $(x_2, y_2)$. Then $x_1x_2 + y_1y_2$ has the value equal to

From an external point $P(h, k)$,a pair of tangent lines are drawn to the parabola $y^2 = 4x$. If $\theta_1$ and $\theta_2$ are the inclinations of these tangents with the $x$-axis such that $\theta_1 + \theta_2 = \frac{\pi}{4}$,then the locus of $P$ is:

If $x + y = k$ is a normal to the parabola ${y^2} = 12x$,then $k$ is

For the parabola $y^2+6y-2x+5=0$:
$(I)$ The vertex is $(-2,-3)$
$(II)$ The directrix is $y+3=0$
Which of the following is correct?