The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and $100 \, m$ long is supported by vertical wires attached to the cable,the longest wire being $30 \, m$ and the shortest being $6 \, m$. Find the length of a supporting wire attached to the roadway $18 \, m$ from the middle. (in $, m$)

The equations $x = \frac{t}{4}$ and $y = \frac{t^2}{4}$ represent:

What is the area of the triangle formed by the vertex and the endpoints of the latus rectum of the parabola $x^2 = 8y$?

The line $x - y + 2 = 0$ touches the parabola $y^2 = 8x$ at the point

Let $P$ and $Q$ be distinct points on the parabola $y^2=2x$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\Delta OPQ$ is $3\sqrt{2}$,then which of the following is (are) the coordinates of $P$?
$(A)$ $(4, 2\sqrt{2})$
$(B)$ $(9, 3\sqrt{2})$
$(C)$ $(\frac{1}{4}, \frac{1}{\sqrt{2}})$
$(D)$ $(1, \sqrt{2})$

The axis of the parabola $9y^2 - 16x - 12y - 57 = 0$ is