Even though you struggle to get along with your grandmother, she still has spent many years being responsible for you, and worrying about you. It is easy to understand that she would be uncomfortable with you going far away.
So, identify a trusted adult (your guidance counselor maybe?) who can help you break the news to your grandmother if you do get into the far-away school. This person should be able to help her understand that this is truly an exceptional opportunity that should not be missed. This person should also be able to help the two of you come up with a plan for how you will stay in touch. Maybe you will agree to skype twice a week. Maybe you will plan now to travel home a certain number of times each year. You also should bring your younger brother into the conversation. Once you leave, he will be alone with Grandma. That may be scary to both/one/neither of them.
When you are communicating with the trusted adult, be prepared to argue your case for the dream school. It truly should be exceptional for what you want to do. If that adult thinks that you have other equivalent options that are not clear across the continent, it will be harder to get the support that you need when speaking with Grandma.