Which of the following represents a parabola?

If the line $x + \alpha y + \beta = 0$ touches the curve $4x^3 + 4y^3 = xy(xy + 16)$ at points $(x_1, y_1)$ and $(x_2, y_2)$,where $x_1 \neq x_2$,then the value of $\alpha + \beta$ is (given $\alpha, \beta \in R$).

$A$ normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, -\alpha)$ to the parabola $x^2 = -4ay$,where $a > 0$. Let $L$ be the line passing through $(0, -\alpha)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points $A$ and $B$. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment $AB$. If $r : s = 1 : 16$,then the value of $24a$ is. . . .

If $x = my + c$ is a normal to the parabola ${x^2} = 4ay$,then the value of $c$ is

The equation of the parabola whose focus is the point $(0, 0)$ and the tangent at the vertex is $x - y + 1 = 0$ is

If $b$ and $c$ are the lengths of the segments of any focal chord of the parabola $y^2 = 4ax$,then what is the length of the semi-latus rectum?