Prove that for a suitable point `P` on the axis of the parabola, chord `A B` through the point `P` can be drawn such that `[(1/(A P^2))+(1/(B P^2))]` is same for all positions of the chord.

Prove that on the axis of any parabola there is a certain point 'k' which has the property that, if a chord PQ of parabola be drawn through it then 1/(PK)^2+1/(QK)^2 is the same for all positions of the chord.

Prove that the locus of the middle points of all chords of the parabola y^2 = 4ax passing through the vertex is the parabola y^2 = 2ax .

A point P on the parabola y^2=12x . A foot of perpendicular from point P is drawn to the axis of parabola is point N. A line passing through mid-point of PN is drawn parallel to the axis interescts the parabola at point Q. The y intercept of the line NQ is 4. Then-

If the point P(4, -2) is the one end of the focal chord PQ of the parabola y^(2)=x, then the slope of the tangent at Q, is

The tangent at a point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through the point (0,-b) and the normal at P passes through the point (2asqrt(2),0) . Then the eccentricity of the hyperbola is

If PQ and Rs are normal chords of the parabola y^(2) = 8x and the points P,Q,R,S are concyclic, then

The normal chord of a parabola y^2= 4ax at the point P(x_1, x_1) subtends a right angle at the

A curve C has the property that if the tangent drawn at any point P on C meets the co-ordinate axis at A and B , then P is the mid-point of A Bdot The curve passes through the point (1,1). Determine the equation of the curve.

The tangent and normal at P(t) , for all real positive t , to the parabola y^2= 4ax meet the axis of the parabola in T and G respectively, then the angle at which the tangent at P to the parabola is inclined to the tangent at P to the circle passing through the points P, T and G is

A variable chord of circle x^(2)+y^(2)+2gx+2fy+c=0 passes through the point P(x_(1),y_(1)) . Find the locus of the midpoint of the chord.