find the equation of parabola , when the vertex is(2,1) and directrix is x=y-1

Equation of directrix= x - y+1=0--------(i)  

Let the coordinate of focus = (a,b)

 the axis of parabola is perpendicular to directrix.

So the equation of axis of parabola may be taken as x+y+k=0

it passes through  (2,1)

⇒ 2+1+k=0

⇒ k=-3

⇒ the equation of axis of parabola is x+y-3=0---------------(ii)

Now, for the point of intersection of directrix and axis of parabola

On solving (i) and (ii), we get,

y = 2

put the value of y in (i), we get, 

x = 1

Thus, the point of intersection of directrix and axis of parabola 

(X, Y) = (1, 2)

Coordinates of 

z = (1,2)

Vertex A (2,1) is the mid-point of focus and 

z = (1,2)

2 = (a+1)/2

a = 3

1 = (b+2)/2

b = 0

Coordinate of focus = (3,0)

Now you can simply get the equation of parabola by

PS = PM