The equation of the tangent at $P(-4, -4)$ on the curve $x^{2} = -4y$ is

Find the axis of the parabola $9y^2 - 16x - 12y - 57 = 0$.

If a normal chord of the parabola $y^2 = 4ax$ subtends a right angle at the vertex,its slope is-

An equilateral triangle is inscribed in the parabola $y^2 = 4ax$ such that one of its vertices is at the origin $(0, 0)$ and the other two vertices lie on the parabola. The length of its side is equal to

For the parabola $y^2 = 8x$,let $\Delta_1$ be the area of the triangle formed by the endpoints of the latus rectum and the point $P \left( \frac{1}{2}, 2 \right)$ on the parabola. Let $\Delta_2$ be the area of the triangle formed by the tangents at the endpoints of the latus rectum and the tangent at point $P$. Find the value of $\frac{\Delta_1}{\Delta_2}$.

Suppose the parabola $(y-k)^2 = 4a(x-h)$ has vertex $A$ and passes through $O = (0,0)$ and $L = (0,2)$. Let $D$ be an end point of the latus rectum. Let the $Y$-axis intersect the axis of the parabola at $P$. Then,$\angle PDA$ is equal to