`A P` is perpendicular to `P B` , where `A` is the vertex of the parabola `y^2=4x` and `P` is on the parabola. `B` is on the axis of the parabola. Then find the locus of the centroid of ` P A Bdot`

Find the vertex of the parabola x^2=2(2x+y) .

Find the vertex of the parabola y = - 2x^(2) + 12 x - 17 .

A square has one vertex at the vertex of the parabola y^2=4a x and the diagonal through the vertex lies along the axis of the parabola. If the ends of the other diagonal lie on the parabola, the coordinates of the vertices of the square are (a) (4a ,4a) (b) (4a ,-4a) (c) (0,0) (d) (8a ,0)

Find the locus of the middle points of the chords of the parabola y^2=4a x which subtend a right angle at the vertex of the parabola.

P(x , y) is a variable point on the parabola y^2=4a x and Q(x+c ,y+c) is another variable point, where c is a constant. The locus of the midpoint of P Q is

Let y=x+1 is axis of parabola, y+x-4=0 is tangent of same parabola at its vertex and y=2x+3 is one of its tangents. Then find the focus of the parabola.

An equilateral triangle is inscribed in the parabola y^(2)=4ax , where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

A parabola of latus rectum l touches a fixed equal parabola. The axes of two parabolas are parallel. Then find the locus of the vertex of the moving parabola.

The tangent at any point P on the circle x^2+y^2=4 meets the coordinate axes at A and B . Then find the locus of the midpoint of A Bdot

Let P be the midpoint of a chord joining the vertex of the parabola y^(2) = 8x to another point on it . Then locus of P .