The distance of the point $(6, -2 \sqrt{2})$ from the common tangent $y = mx + c$ $(m > 0)$ of the curves $x = 2y^2$ and $x = 1 + y^2$ is

If $L$ is the normal drawn to the parabola $y^2 = 8x$ at the point $t = \frac{1}{\sqrt{2}}$,then the foot of the perpendicular drawn from the focus of the parabola on to the normal $L$ is

The length of the latus rectum of the parabola $9x^2 + 16y^2 + 24xy - 4x + 3y = 0$ is

Let $P(4, -4)$ and $Q(9, 6)$ be two points on the parabola $y^2 = 4x$. Let $X$ be any point on the arc $POQ$ of this parabola,where $O$ is the vertex,such that the area of $\Delta PXQ$ is maximum. Then this maximum area (in sq. units) is

If $2x + y + \lambda = 0$ is a focal chord of the parabola $y^2 = -8x$,find the value of $\lambda$.

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 $Q^{\prime}$