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

We have the equation of the tangent at the vertex equal to
x + y = 1.

x + y = 1

=>
y = -x + 1

The slope of  the tangent is -1, the slope of
the line perpendicular to the tangent is 1. This line passes through the focus. y = x +
c passes through (1,1). This gives the equation of the line as y =
x

The point of intersection of x + y = 1 and x = y can be
got by

y + y = 1

=> 2y
= 1

=> y = 1/2

The
vertex is (1/2, 1/2)

The distance between the focus and
vertex is sqrt [2*(1/2)^2] = sqrt 2 / 2

The equation of the
parabola is (y - 1/2)^2 = (2*sqrt 2 ) * ( x - 1/2)y^2 + 1/4 - y = 2*sqrt 2*x - sqrt
2

=> 4y^2 - 4y - 8*sqrt 2*x + 4*sqrt 2 + 1 =
0

The required equation of the
parabola is 4y^2 - 4y - 8*sqrt 2*x + 4*sqrt 2 + 1 =
0