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