Let the equation of the tangent at a point $P$ on the parabola $x^2-4x-4y+16=0$ be $2x-y-5=0$. If the equation of the normal drawn at $P$ to this parabola is $ax+y+c=0$,then find the value of $ac$.

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

What is the length of the latus rectum of the parabola $x^2 = -y$?

Normals at $P, Q, R$ are drawn to the parabola $y^2 = 4x$ which intersect at the point $(3, 0)$. Then,$\Delta PQR$ is:

Let $E$ denote the parabola $y^2=8x$. Let $P=(-2,4)$,and let $Q$ and $Q^{\prime}$ be two distinct points on $E$ such that the lines $PQ$ and $PQ^{\prime}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) $TRUE$?
$(A)$ The triangle $PFQ$ is a right-angled triangle
$(B)$ The triangle $QPQ^{\prime}$ is a right-angled triangle
$(C)$ The distance between $P$ and $F$ is $5\sqrt{2}$
$(D)$ $F$ lies on the line joining $Q$ and $Q^{\prime}$

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