Ok,

$\displaystyle \frac{dy}{dx} = \frac{2 - y^2}{2xy - 3}$

1. Show that curve has no tangent parallel to y axis.

Well, when thats the case, dy/dx = 0.

Giving

$\displaystyle 2 - y^2 = 0$

And therefore $\displaystyle y = +-/sqrt{2}$

Well you were meant to do

2xy - 3 = 0 and sub into original equation to show there is no solution.

And my question is Why? Do you always do 1/(dy/dx) and equate that to 0 to find a line parallel to y axis? Why do you do that?