Find the equation of the tangent at the vertex of the parabola $4y^2 + 6x = 8y + 7$.

The area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12y$ to the extremities of its latus rectum is

If a perpendicular drawn through the vertex $O$ of the parabola $y^2=4ax$ to any of its tangent meets the tangent at $N$ and the parabola at $M$,then $ON \cdot OM=$

If the normal at $(ap^2, 2ap)$ on the parabola $y^2 = 4ax$ meets the parabola again at $(aq^2, 2aq)$,then:

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 $\frac{\Delta_1}{\Delta_2}$.

The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4x$ is