`M` is the foot of the perpendicular from a point `P` on a parabola `y^2=4a x` to its directrix and `S P M` is an equilateral triangle, where S is the focus. Then find `S P` .

If P be a point on the parabola y^2=3(2x-3) and M is the foot of perpendicular drawn from the point P on the directrix of the parabola, then find length of each sides of an equilateral triangle SMP(where S is the focus of the parabola).

If the tangent at any point P to the parabola y^(2)=4ax meets the directrix at the point K , then the angle which KP subtends at its focus is-

Let N be the foot of perpendicular to the x-axis from point P on the parabola y^2=4a xdot A straight line is drawn parallel to the axis which bisects P N and cuts the curve at Q ; if N Q meets the tangent at the vertex at a point then prove that A T=2/3P Ndot

The coordinates of the foot of the perpendicular drawn from the point P(x,y,z) upon the zx- plane are-

The tangent at any point P onthe parabola y^2=4a x intersects the y-axis at Qdot Then tangent to the circumcircle of triangle P Q S(S is the focus) at Q is

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

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

If P N is the perpendicular from a point on a rectangular hyperbola x y=c^2 to its asymptotes, then find the locus of the midpoint of P N

Find the points of contact Q and R of a tangent from the point P(2,3) on the parabola y^2=4xdot

A B C is an equilateral triangle with A(0,0) and B(a ,0) , (a>0). L, M and N are the foot of the perpendiculars drawn from a point P to the side A B ,B C ,a n dC A , respectively. If P lies inside the triangle and satisfies the condition P L^2=P MdotP N , then find the locus of Pdot