Given:
- Directrix of the parabola: x + y - 2 = 0
- Line touching the parabola: 2x + y - 5 = 0
- Point of tangency: (2, 1)

To find:
- Semi-latus rectum of the parabola

Solution:

Step 1: Find the equation of the parabola
The equation of a parabola with a vertical directrix is given by:
(y - k)^2 = 4a(x - h)

The given directrix equation is x + y - 2 = 0. We rearrange it to the standard form:
x = -y + 2

Comparing this with the general equation, we have:
h = 0
k = 2

Substituting the values of h and k, the equation of the parabola becomes:
(y - 2)^2 = 4a(x - 0)
(y - 2)^2 = 4ax^2

Step 2: Find the slope of the line
The given line equation is 2x + y - 5 = 0. We rearrange it to the slope-intercept form:
y = -2x + 5

Comparing this with the general form y = mx + c, we have:
m = -2

Step 3: Find the slope of the tangent at the point of tangency
The tangent at the point (2, 1) will have the same slope as the line it touches. Therefore, the slope of the tangent is also -2.

Step 4: Find the vertex of the parabola
The vertex of the parabola is the point of intersection of the directrix and the line perpendicular to the directrix passing through the point of tangency.

Since the directrix is x + y - 2 = 0, the perpendicular line will have the slope 1.

Using the point-slope form, the equation of the perpendicular line passing through (2, 1) is:
y - 1 = 1(x - 2)
y - 1 = x - 2
y = x - 1

Solving the equations x = -y + 2 and y = x - 1, we find the vertex of the parabola to be (1, -1).

Step 5: Find the distance between the vertex and the point of tangency
Using the distance formula, the distance between (1, -1) and (2, 1) is:
d = sqrt((2 - 1)^2 + (1 - (-1))^2)
d = sqrt(1 + 4)
d = sqrt(5)

Step 6: Find the semi-latus rectum
The semi-latus rectum of a parabola is given by the formula:
l = 2a

Using the distance between the vertex and the point of tangency, we have:
2a = sqrt(5)
a = sqrt(5) / 2

Therefore, the semi-latus rectum of the parabola is sqrt(5) / 2.

9/√2 but m not sure