The simplest equation (standard form) that would solve the problem is

 y^2=4ax.  

This parabola has its axis on the x-axis with latus rectum of 4a.

However this can be translated around the plane, so allowing for constants b and c, we can generalize to get

(y-b)^2=4a(x-c). 

Taking differentials of both sides and solving for dy/dx one gets 

dy/dx = 2a/(y-b) which I believe solves the problem.

The only other generalization that can be made is to add a constant D such that

D(y-b)^2=4a(x-c)  but when you put it into standard form you get

(y-b)^2=4a/D * (x-c) 

so the latus rectum length is now 4a/D.  So I will go out on a limb and say the solution I have above is the answer.  However, I have been wrong before, LOL.