If two perpendicular tangents are drawn from a point $P$ to the parabola $y^2 = 4x$,find the locus of $P$.

Let $P$ be the mid-point of a chord joining the vertex of the parabola $y^{2}=8x$ to another point on it. Then,the locus of $P$ is

The equation of the tangent to the parabola $y = (x - 3)^2$ parallel to the chord joining the points $(3, 0)$ and $(4, 1)$ is:

The equation of the parabola with focus $(0,0)$ and directrix $x+y=4$ is

An equilateral triangle is inscribed in the parabola $y^2 = 8x$,with one of its vertices at the vertex of the parabola. Then,the length of the side of that triangle is

Find the equation of the common tangent to the parabolas $y = x^2$ and $y = -(x - 2)^2$.