Well, I guess it all depends on where you live. I live in California. Being that your license plate isn't 3 feet long, can I assume that you're from the states? West coast? Upper east coast? Possibly 🇨🇦?
In California, if you get a ticket, it goes down as a "point" on your driving record and your insurance company automatically gets notified. Now, if you go to traffic school after you get a ticket, the DMV doesn't put that "point" on your driving record, therefore, you're insurance rates won't go up.
As for fighting the ticket.... the police officer wrote you a citation for "following too closely"... now, how would that fat sum-a-bitch know if you were following too close? He wasn't even there?! I mean, sure, you got "close enough" to make contact with her vehicle, but that's neither here nor there. Maybe "her break lights weren't working"? Maybe "you had an allergic reaction to some shell fish and you pasted out at the wheel"? MAYBE "a water bottle rolled on to the floor board and under your break pedal, therefore, making it impossible to depress said break pedal."
Over here there is a company called "Ticket Clinic". I've never personally used their services, but my friend swears by them. He said that he only had to pay $200 for a ticket that would have otherwise cost $1000. He said that you don't even have to go to court, they go to court on your behalf...maybe something to look in to.