`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 Q is the foot of the perpendicular from a point p on the parabola y^(2)=8(x-3) to its directrix. S is an equilateral triangle then find the lengh of side of the triangle.

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).

A point P on the parabola y^(2)=4x , the foot of the perpendicular from it upon the directrix and the focus are the vertices of an equilateral triangle. If the area of the equilateral triangle is beta sq. units, then the value of beta^(2) is

If a point P on y^2x , the foot of the perpendicular from P on the directrix and the focus form an equilateral traingle , then the coordinates of P may be

A point on a parabola y^2=4a x , the foot of the perpendicular from it upon the directrix, and the focus are the vertices of an equilateral triangle. The focal distance of the point is equal to a/2 (b) a (c) 2a (d) 4a

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

S is the point (0,4) and M is the foot of the perpendicular drawn from a point P to the y -axis. If P moves such that the distance PS and PM remain equal find the locus of P .

Show that length of perpendecular from focus 'S' of the parabola y^(2)= 4ax on the Tangent at P is sqrt(OS.SP)

The co-ordinates of the point S are (4, 0) and a point P has coordinates (x, y). Express PS^(2) in terms of x and y. Given that M is the foot of the perpendicular from P to the y-axis and that the point P moves so that lengths PS and PM are equal, prove that the locus of P is 8x=y^(2)+16 . Find the co-ordinates of one of the two points on the curve whose distance from S is 20 units.

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