demand curves from utility functions! HELP?

Question

Derive the demand curve for the utility function U= X^2+Y^2. Calculate own price and cross price elasticity. Evaluate these elasticities at M=12, PX=3 and PY=2

Answer 1

I assume the budget constraint is M=Px X+Py Y.
Then the maximum U subject to the constraint is at the point where the utility curve is tangent to the budget constraint.
that is dY/dX=-Px/Py
and dU/dX=2X and dU/dY=2Y so dY/dX=Y/X
so Y/X=-Px/Py or Px X=-Py Y
(If you have been taught how to you Lagrangian multipliers you should use them to get the answer instead of this non-rigorous method)
Now you can find the quantities by subtitution into the equation for M.