find the points at which the graph of 4x^2 - 4x + 12y^2 -6y = 10 has a vertical and horizontal tangent line.

For the vertical only solve for y and for the horizontal only solve for x.

I got the deriviative to be:
(8x-4)/(-24y + 6).

For the horizontal, x = 1/2
For the vertical y = 1/4

I have to plug these into the original equation to get the points and solve for the rest of the equation but each time i do it, it doesn't turn out right.
Can someone check over my work and show me how to finish the problem?

Thanks