We have an underground line from the meter to our house. We have added a new
Living area. Can we dig down and split the main line to come into the house at a different location for the new addition?

Not a good idea. You need to have an electrician add another breaker or two into your existing main breaker panel and have the new addition wired according to code into the main breaker panel. The way you were thinking of doing this could cause you to be electrocuted or burn your home down if improperly wired. Your addition will have to be properly inspected and any building or wiring will have to be to code. The added expense of an electrician is well worth it. You paid to have a permit for your new addition I hope?

Even though I only had a 10X12 shed that I wanted to have a ceiling light, light switch, and 2 outlets added, I hired an electrician to do the job. He added two extra breakers to my existing main panel. I can run two different power tools at once using both recepticles at the same time without having the breakers snap as he wired them for a higher amp use, i.e. an electric saw and a drill. I discussed my needs with him before he did this job. It only cost me $200 and was able to pass code.

Absolutely Not! The NEC only allows a single point of entry into a residence.

Add to that the Power company owns the service equipment including the meter from the meter to the power company.

What you can do is is run a raceway from the main panel to a sub panel to feed the sub panel and then distribute the electricity through the sub panel via additional branch circuits.