If the locus of a point that divides a chord of slope 2 of the parabola $y^2=4 x$ internally in the ratio $1: 2$ is a parabola, then its vertex is

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15 x$ subtends an angle $\theta$ at the vertex of the parabola, then $\sin \frac{\theta}{3}+\cos \frac{2 \theta}{3}-\sec \frac{4 \theta}{3}=$

Tangents are drawn at three points $P\left(t_1\right), Q\left(t_2\right), R\left(t_3\right)$ on the parabola $y^2=x$. Let these tangents intersect each other at the points $L, M, N$. If $t_1=2, t_2=-4, t_3=6$, then the area of the $\triangle L M N$ is

If the tangents of the parabola $y^2=8 x$ passing through the point $P(1,3)$ touches the parabola at $A$ and $B$, then the area (in sq. units) of $\triangle P A B$ is