Homework Help: The Shortest Timed Path

1. The problem statement, all variables and given/known data
A frictionless path is set up between two points in space at different heights, such that a body released from the higher point along this path takes minimum possible time to reach the lower point. Show that the path is a cycloid. Also show that a body released along this path takes about 25% lesser time as compared to travelling along a straight path.
2. Relevant equations
Energy conservation,


3. The attempt at a solution

Let us assume that the total height difference is H and the horizontal distance is X.
when the body falls through a height y its speed is [tex]\sqrt{2gy}[/tex]
[tex]\frac{dx}{dt}[/tex]= vcos[tex]\theta[/tex]
[tex]\frac{dy}{dt}[/tex]= vsin[tex]\theta[/tex]
i am unable to integrate and apply the condition for minimum time- please help.