`L L '` is the latus sectum of the parabola `y^2=4a xa n dP P '` is a double ordinate drawn between the vertex and the latus rectum. Show that the area of the trapezium `P P^(prime)L L '` is maximum when the distance `P P '` from the vertex is `a//9.`

If the segment intercepted by the parabola y^2=4a x with the line l x+m y+n=0 subtends a right angle at the vertex, then

Through the vertex O of the parabola y^(2) = 4ax , a perpendicular is drawn to any tangent meeting it at P and the parabola at Q. Then OP, 2a and OQ are in

The tangents to the parabola y^2=4a x at the vertex V and any point P meet at Q . If S is the focus, then prove that S P,S Q , and S V are in GP.

If a tangent to the parabola y^2=4a x meets the x-axis at T and intersects the tangents at vertex A at P , and rectangle T A P Q is completed, then find the locus of point Qdot

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 electric field at a point P on the perpendicular bisector of a uniformly charged rod. The lengthof the rod is L, the charge on it is Q and the distance of P from the centre of the rod is a.

PQ is a normal chord of the parabola y^2= 4ax at P,A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of AR is : (A) equal to the length of the latus rectum (B) equal to the focal distance of the point P (C) equal to the twice of the focal distance of the point P (D) equal to the distance of the point P from the directrix.

Statement 1: If the straight line x=8 meets the parabola y^2=8x at Pa n dQ , then P Q substends a right angle at the origin. Statement 2: Double ordinate equal to twice of latus rectum of a parabola subtends a right angle at the vertex.

A tangent is drawn to the parabola y^2=4 x at the point P whose abscissa lies in the interval (1, 4). The maximum possible area of the triangle formed by the tangent at P , the ordinates of the point P , and the x-axis is equal to

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