If two tangents drawn from a point $P$ to the parabola $y^2 = 4x$ are at right angles,then the locus of $P$ is:

Let $R$ be the focus of the parabola $y^2=20x$ and the line $y=mx+c$ intersect the parabola at two points $P$ and $Q$. Let the point $G(10, 10)$ be the centroid of the triangle $PQR$. If $c-m=6$,then $(PQ)^2$ is

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^{2}=8x$, with one of its vertices at the vertex of this parabola, is (in $\sqrt{3}$)

The length of the latus rectum of the parabola $x^{2}+2y=8x-7$ is

From an external point $P(h, k)$,a pair of tangent lines are drawn to the parabola $y^2 = 4x$. If $\theta_1$ and $\theta_2$ are the inclinations of these tangents with the $x$-axis such that $\theta_1 + \theta_2 = \frac{\pi}{4}$,then the locus of $P$ is:

If the tangent to the curve $x = at^2, y = 2at$ is perpendicular to the $x$-axis,then what is the point of contact?