What would be a geometric Euclidean interpretation of the homogeneous plane (0,0,0,8)?

Question

Homogeneous coordinates have one dimension more than the corresponding Euclidean coordinates. The Euclidean origin can be described with projective coordinates as (0,0,0,1). So, geometrically, what would be an interpretation of the homogeneous coordinates (0,0,0,8) ? This is clearly a point in the Euclidean space, but how does it relate to the origin? What I am trying to say is: How does the last component of homogeneous coordinates relate to Euclidean coordinates?