If the line $y = -x + k$ is normal to the curve $y^2 = 16x$,then $k$ is

The point on the parabola $y^2 = 8x$ at which the normal is inclined at $60^\circ$ to the $x$-axis has the coordinates:

Tangents are drawn at points $(x_i, y_i); i = 1, 2, 3$ lying on the parabola $y = x^2$ to enclose a triangle of area $\Delta$. If $x_1, x_2, x_3$ form an increasing arithmetic progression,where $x_1 = -1$ and $y_3 = 9$,then $\Delta$ is ............. $sq. \, units$.

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$,such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $R$ denote the region lying in the first quadrant,enclosed by the parabola $y^2=x$,the curve $S$,and the lines $x=1$ and $x=4$. Then which of the following statements is (are) True?
$(A) \ (4, \sqrt{3}) \in S$
$(B) \ (5, \sqrt{2}) \in S$
$(C)$ Area of $R$ is $\frac{14}{3}-2 \sqrt{3}$
$(D)$ Area of $R$ is $\frac{14}{3}-\sqrt{3}$

The curve described parametrically by $x = t^2 - 2t + 2$ and $y = t^2 + 2t + 2$ represents:

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 $, m$)