Stop using such an ancient measurement system. from current measurements, you're 84lbs, and that is severely underweight.
Please clarify using an actual measurement system.

Then what are you asking? If you are 84lbs, then you need to gain weight. If you're not providing accurate details then who do you expect to help you?

Honey you need to tell your parents. They won't be angry, they'll be worried.

you really do need to gain weight, at 6 stone your BMI (body mass index) is 14.4 (this is a measurement of your weight in relation to your height, now it really should be between 17 and 25)

and talking to your mum about it may make her worry in the short term but if you left it longer then it would only be worse

You currently have a very unhealthy relationship with food/exercise and probably a dysorphic perception of your own body. You need to understand this can be very dangerous if you ignore it. You need to speak to someone right now. Your parents, an uncle/aunty, or even a teacher. Be brave. They can get you the help you need.

If you continue down this path, you are going to become very very sick. Do you really think your family and friends would want that to happen to you?

If your worried about your moms feelings then don't tell your mom, but you have to tell someone.

Have fun dying of malnutrition, I hear it's great.

You currently have a very harmful connection with food or exercise and probably a isomorphic understanding of your own system. You need to comprehend this can be very risky if you neglect it. You need to talk with someone right now. Your mother and father, an uncle/aunty, or even a instructor. Be courageous. They can get you the help you need.........

I`m 5'4 and I normally weigh 115-117 pounds and I`m a pretty fit guy. Just work out more and eat 1500 calories a day.

Feeling a little messed up about the way you look is pretty normal when you're growing up, you're not alone in the way you feel. Everybody feels like that at some point and it's important to realise that if you get help all these problems may seem silly 2 months from now.

Your Mother undoubtably loves you to bits and I think eventually you'll need to tell her how you're feeling, she'll appreciate it and you'll not have to feel like you're hiding something.
In saying that I get why you don't want to tell her now. Please be aware that what you've described is not a healthy relationship with food and that the way you're feeling is probably making it hard to look at your situation objectively.

Here are things you need to change
Dislike of eating food
Binge eating
Vomiting
Feeling like you need to excercise to work off excess food.

This is typical behaviour of someone who is bulemic.

Here are some good things
You love cooking food
You're a horse rider
You're a dancer
You're known for being skinny

Clearly you're not overweight, or in need to loose weight. You're a horse riding dancer who can cook, that's cool. Please acknowledge that not everything is bad. My suggestion is that you write how you feel on paper so that it's all clear and externalised, it'll make it that much easier for the person you talk to to understand you. Then find someone to talk to, aunt, uncle, teacher, etc. Most schools have a councellor you can speak with.

Chances are that after talking to someone you'll realise that your problems aren't as big as you think they are. How you're feeling now will not matter in 1,2,5 years time if you let someone help.