if the focus of a parabola is (-2,1) and equation of the directrix is x+y=3, find the vertex of the parabola

vertex of the parabola is the mid-point of the line segment joining the focus to the intersection point of axis and directrix.
let B be the intersection point of axis and directrix.
then B is the foot of perpendicular drawn from the focus to the directrix.
let the coordinates of B be (h,k).
since axis is perpendicular to directrix.
the equation of directrix is y = -x+3, slope of directrix =-1
slope of axis = 1.
equation of line passing through the point (-2,1) and having slope 1, is
$$\begin{gathered} y-1=1*(x+2) \\ x-y=-3 \\ \sin ce (h,k) lies on axis, therefore \\ h-k=-3 ..........\left(1\right) \\ (h,k) also lies on directrix, therefore \\ h+k=3 ........\left(2\right) \\ on solving eq\left(1\right) and eq\left(2\right), we have: \\ 2h=0\Rightarrow h=0 \\ and k=3 \end{gathered}$$
the coordinates of vertex of parabola is (0,3).

hope this helps you