Length of a Perpendicular Line between two lines

Let AB represent the given line xcosa + ysina = p.

Let N' (x1, y1) be the given point.

Through N', draw B'N'A'|| BA, so that OM produced will meet B'A' at N.

Then, MN = M'N' = p'.

ON = OM + MN = p + p'

The equation of the line A'N'B' is xcosa + ysina = p + p'

The point N'(x₁, y₁) lies on A'B'.