Differentiate

A very peculiar exercise since by definition (dx/dy)*(dy/dx) is equal to 1!?
You simply differentiate the equation noting that d(x²)= 2x (similarly for y²) and that 2hxy is 2hydx and 2hxdy and that c² becomes 0.
Then form from this equation dy/dx and dx/dy and you'll see that when multiplied they give 1.

Where (dx/dy)*(dy/dx) does not equal 1 is when either dx/dy or dy/dx does not exist. Other that in specific cases of functions at certain points (like 1/x at x=0) the Weierstrass non-differentiable  function's are the best examples.
http://perso-math.univ-mlv.fr/users/jaffard.stephane/pdf/Hardy.pdf

You have to look at the family of curves that are dependent upon a, b, c, and h.
For example:
x² - 2xy + y² = 1 gives two parallel lines: https://www.wolframalpha.com/input/?i=x%5E2+-+2xy+%2B++y%5E2+%3D+1

You can also get circle, ellipses, and hyperbolas from your equation.

For the circle, dy/dx = -x/y, but what if y=0? The tangent line to the circle is vertical, so the derivative is infinity.
https://www.physicsforums.com/threads/differntiating-a-circle.279719/

Here is a brief video that covers one of your cases, and covers that case where the slope is undefined.
https://www.youtube.com/watch?v=ua6ZTac8d0w