The ends of the latus rectum of the conic ${x^2} + 10x - 16y + 25 = 0$ are

The area of the triangle formed by the lines joining the vertex of the parabola,$x^2 = 8y$,to the extremities of its latus rectum is

What is the equation of the directrix of the parabola $y^2 + 4y + 4x + 2 = 0$?

Let the focal chord $PQ$ of the parabola $y^2=4x$ make an angle of $60^{\circ}$ with the positive $x$-axis,where $P$ lies in the first quadrant. If the circle,whose one diameter is $PS$,$S$ being the focus of the parabola,touches the $y$-axis at the point $(0, \alpha)$,then $5 \alpha^2$ is equal to :

The intercepts of the focal chord (which is a part of the latus rectum) to the parabola $y^{2}=4ax$ are $b$ and $k$. Then $k$ is equal to:

Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y =-\frac{1}{2}$. Then $S=\left\{x \in R : \tan ^{-1}\left(\sqrt{f(x)}+\sin ^{-1}(\sqrt{f(x)+1})\right)=\frac{\pi}{2}\right\}$: