How do I show that dy/dx does not equal zero?

Question

The equation of a curve is $2x^2-3xy+y^2=5$.

1. Find the equations of the tangent and normal to the curve at point $(4,3)$.
2. Show that there is no point on the curve at which the tangent is parallel to the x-axis.
3. Show that every line parallel to the x-axis cuts the curve at two distinct points.

For part 1, I was able to get the answer. however, for part 2, I was stuck. I thought of trying to show that $dy/dx$ cannot equals to zero. But how do I do that?