find the equation of parabola whose focus is (1,1) and tangent at the vertex is x + y = 1

let S be the focus and A be the vertex of the parabola. let K be the point of intersection of the axis and directrix.

since axis is a line passing through S(1,1) and perpendicular to x+y=1. 

so, let the equation of the axis be x-y+λ=0

this will pass through (1,1) if 1-1+λ=0 ⇒λ=0

so the equation of the axis is x-y=0

the vertex A is the point of intersection of x-y=0 and x+y=1. solving these two equations, we get x=1/2 , y=1/2

so, the coordinates of the vertex A are (1/2,1/2).

let be the coordinates of K. then,

so, the co-ordinates of K are (0,0).

since directrix is a line passing through K(0,0) and parallel to x+y=1.

therefore, equation of the directrix is y-0=-1(x-0), i.e. x+y=0

let P(x,y) be any point on the parabola. then,

distance of P from the focus S= distance of P from the directrix

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