The locus of the point of intersection of the tangents drawn at the ends of a focal chord of the parabola $x^{2}=-8y$ is

If $(x_1, y_1)$ and $(x_2, y_2)$ are the points on the parabola $y^2 = 32x$ each at a focal distance of $10$ units,then $2(x_1^2 + x_2^2 + y_1^2 + y_2^2) = $

For what value of $k$ does the line $2y - x + k = 0$ touch the parabola $x^2 + 4y = 0$?

Let $A(0,1)$,$B(1,1)$,and $C(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y^2 = 4ax$ passing through $D$ is $(\alpha + \beta \sqrt{2}, 0)$,where $\alpha$ and $\beta$ are rational numbers,then $\frac{\alpha}{\beta^2}$ is equal to

The vertex of the parabola $(y - 2)^2 = 16(x - 1)$ is

The equation of the normals at the ends of the latus rectum of the parabola $y^{2} = 4ax$ is: