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

The locus of the midpoint of the line segment joining the focus to a moving point on the parabola $y^2 = 4ax$ is another parabola with the directrix

Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $y^{2}=12x$.

If $P$ and the origin are the points of intersection of the parabolas $y^2=32x$ and $2x^2=27y$,and if $\theta$ is the acute angle between these curves at $P$,then $5\sqrt{\tan \theta} =$

The normal at a point on the parabola $y^2 = 4x$ passes through a point $P$. Two more normals to this parabola also pass through $P$. If the centroid of the triangle formed by the feet of these three normals is $G(2,0)$,then the abscissa of $P$ is

Find the measure of the angle in degrees subtended by the double ordinate of the parabola $y^2 = 4ax$ at its vertex.