Prove that the equation of the parabola whose focus is (0, 0) and tangent at the vertex is `x-y+1 = 0 `is `x^2 + y^2 + 2xy - 4x + 4y - 4=0`.

Consider the parabola whose focus is at (0,0) and tangent at vertex is x-y+1=0 The length of latus rectum is

The vertex of the parabola y^2 + 4x = 0 is

Find the equation of parabola whose focus is (0,1) and the directrix is x+2=0. Also find the vertex of the parabola.

Find the length of the latus rectum of the parabola whose focus is at (2, 3) and directrix is the line x-4y+3=0 .

The vertex of the parabola y^2 = 4x + 4y is

Consider the parabola whose focus is at (0,0) and tangent at vertex is x-y+1=0 Tangents drawn to the parabola at the extremities of the chord 3x+2y=0 intersect at angle

Find the points on the parabola y^2-2y-4x=0 whose focal length is 6.

Find the equation of the parabola whose focus is S(-1,1) and directrix is 4x+3y-24=0 . Also find its axis, the vertex, the length, and the equation of the latus rectum.

Prove that the equation of any tangent to the circle x^2+y^2-2x+4y-4=0 is of the form y=m(x-1)+3sqrt(1+m^2)-2.