You don't say why your mother refused. Does she disapprove of dancing, or is it the cost of lessons, or something else? As an adult, I have to say, it's probably not a good idea to lie to your mom about dancing. Chances are, she'll find out and it will hurt her trust in you.
That said, here are some ideas for getting into back into dance:
--does your school have any dance classes or after-school activities in dance?
-- check out your local parks, recreation dept., YMCA, community colleges, etc. to see if there are low-cost classes you can take
-- sometimes a dance studio will let you work in exchange for classes or lessons
If you present a specific plan to your mom--say, a class that meets for a month or so--she might be willing to let you take it with the idea that you will both re-consider your continuing in dance when the class is over. If she sees you sticking with the class, being happy, and getting all your schoolwork and other responsibilities done, she might change her mind.
If the worst happens, and she continues to refuse, then keep watching dance and start an exercise program of aerobics, strength, and stretching, so your body will be ready to move ahead fast when you turn 18 and can make your own decisions. If you're going to college, check out whether your choices have ballroom dance clubs or teams (lots of colleges do). If you're not going to college, check out becoming a dance instructor; lots of studios have training programs and will take trainees with little or no experience if they have an aptitude for dance.
Good luck! Let us know how it goes!