The equation of a hyperbola is
$ x^2 - y^2 = r $
Assuming $ a = b = 0 $
Suppose now the equation
$ x^2 - y^2 + z^2 = r $
What kind of curve would this equation parametrize ? What is its geometry?
Since $ x^2 + y^2 = r $ is the equation of a circle would this be like a mix between a hyperbola (in the xy plane) and a circle (in the yz plane) ?
How does a variation in r change the shape of this curve?
Suppose we scale the variables by some factors
$ x^2 - ky^2 + mz^2 = r $
How would this factors affect the shape?
Take the surface $$x^2-y^2+z^2=r,$$ for some $r>0$.
If we project onto a plane $x=c$, we get $$-y^2+z^2=r-c^2,$$ a hyperbola (depending on the sign of $r-c^2$ we may get a rotated hyperbola).
Doing the same with $y=c$, we get $$x^2+z^2=r+c^2,$$ which is always a circumference.
If we project onto a plane $z=c$, we get $$x^2-y^2=r-c^2,$$ that is, a hyperbola again.
The result is a one-sheet hyperboloid (I suggest you use software to graph the surface and all the previous projections)
In case $r$ was $0$ we would get a cone and if $r<0$ we would get a two-sheet hyperboloid (for some $c$ we wouldn't get circumferences in the second projection).
For your last enquiry, I will asume $k,m>0$ and will leave other cases for you as an exercise. You just need to notice that the change of variables $$x'=x, y'=\sqrt{k}y, z'=\sqrt{m}z$$ reduces the equation to the previously studied case (if you went the 'projections route' the only difference would be that you would get ellipses instead of circumferences).