At any point P on the parabola `y^2 -2y-4x+5=0` a tangent is drawn which meets the directrix at Q. Find the locus of point R which divides QP externally in the ratio `1/2:1`

Let P be a point on the parabola y^(2) - 2y - 4x+5=0 , such that the tangent on the parabola at P intersects the directrix at point Q. Let R be the point that divides the line segment PQ externally in the ratio 1/2 : 1. Find the locus of R.

The ordinates of points P and Q on the parabola y^2=12x are in the ration 1:2 . Find the locus of the point of intersection of the normals to the parabola at P and Q.

Find the coordinates of the point R which divides PQ externally in the ratio 2:1 and verify that Q is the mid point of PR.

Tangents are drawn from the point (-1,2) to the parabola y^(2)=4x. These tangents meet the line x=2 at P and Q, then length of PQ is

y=2x+c, 'c' being variable is a chord of the parabola y^2=4x , meeting the parabola at A and B. Locus of a point dividing the segment AB internally in the ratio 1 : 1 is

the tangent drawn at any point P to the parabola y^(2)=4ax meets the directrix at the point K. Then the angle which KP subtends at the focus is

The tangent at any point P on y^(2)=4x meets x-axis at Q, then locus of mid point of PQ will be

Two tangents to the parabola y^(2) = 8x meet the tangent at its vertex in the points P & Q. If PQ = 4 units, prove that the locus of the point of the intersection of the two tangents is y^(2) = 8 (x + 2) .