Posted by anu on Friday, January 7, 2011 at 1:37am.
A. a particle moves around the circle x^2+y^2 = 1 in such a way that the x coord rate of change is dx/dy= y
B) IS THE PARTICLE MOVING COLCIKWISER OR COUNTERCLOCKWISE AROUND THE CIRCLE
C) FROM THE RT TRIANGLE ONE CAN SEE THAT SIN(THETA) = Y AND COS(THETA) = X
USE THESE RELATIONS TO SHOW THAT THE ANGLE THETA IS CHANGING AT A CONST. RATE AND THEN FIND THAT RATE

calculus  drwls, Friday, January 7, 2011 at 9:28am
B. Something is fishy about this queation.
dx/dy is negative in the first quadrant and third quadrant. y is positive in the first and second quadrant. Furthermore,
dx = y dy means that
x = (y^2/2) + C
where C is any constant.
The particle is not following the circle at all. It is follwing a parabola. Nothing can be said about the direction or rate it is moving because time does not appear in your equations.
Are you sure you copied the problem correctly?
Should you have written dy/dt = x ? 
calculus  Reiny, Friday, January 7, 2011 at 9:33am
agree with drwls
I started working on this, following the precise information as you stated it
x^2 + y^2 = 1
2x dx/dy + 2y dy/dy = 0
dx/dy = y/x , but you said dx/dy = y
y = y/x
x = 1
at this point I stopped.