For the parabola $y^2 = x$,let $PQ$ be a chord such that one endpoint $P$ is $(4, -2)$ and the chord is perpendicular to the axis of the parabola. What is the slope of the normal at $Q$?

For the curve $y^2 = 4ax$,the lengths of the tangent,subtangent,normal,and subnormal at the point $(at^2, 2at)$ are respectively:

The parametric equations of the parabola $y^2-4x-8y-12=0$ are

Find the equation of the normal to the curve $x^{2}=4y$ which passes through the point $(1,2)$.

Consider the parabola $y^2=8x$. Let $\Delta_1$ be the area of the triangle formed by the endpoints of its latus rectum and the point $P\left(\frac{1}{2}, 2\right)$ on the parabola,and $\Delta_2$ be the area of the triangle formed by the intersection points of the tangents drawn at $P$ and at the endpoints of the latus rectum. Then $\frac{\Delta_1}{\Delta_2}$ is

The equation of the common tangent touching the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $x$-axis is