A road cannot move; the road isn't going home (as in "the road is moving to the house itself"). The subject for "가-" is omitted. Here, "집으로 가는 길" literally translates to "the road along which one goes home".
I'm keeping in mind that these spelled and sound alike grammar descriptive verbal parts are not the particles we use directly on nouns in the subject object topic lessons.
"the road going to the house" is the correct answer, but not "the way going..." or the more natural "the road leading to the house"?
and when i skip answering i get "the road home" as correct answer?
If it sounds silly, it’s probably not the right answer. “The home-going road” is already quite a stretch. You’d be hard-pressed to find a use for the expression as-is.