Securing a large screw hook in drywall without an anchor
Cut a 1"x 4" board just long enough to fit between the wall on the right and garage door on the left. 

Attach each end with 2 1/2" screws into the studs of said wall and door frame. 

Screw hook, ( or hooks ) into the board. 

Other wise like i said in the comments, take it to the store to size and anchor, i have seen ones big enough for that size hook.

Be aware that an anchor may loosen up over time as well from repeated hanging and removing of the things suspended on it.