Suppose a parabola $y=ax^2+bx+c$ has two $x$-intercepts,one positive and one negative,and its vertex is $(2,-2)$. Then,which of the following is true?

If $(2, -8)$ is one endpoint of a focal chord of the parabola $y^{2} = 32x$,what is the other endpoint?

The equation of the line touching both parabolas $y^2=4x$ and $x^2=-32y$ is

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

If the normals at two points on the parabola $y^2 = 4ax$ meet on the parabola,then what is the product of the ordinates of these two points?

The length of the subnormal to the parabola $y^2 = 4ax$ at any point is equal to