What kind of curve does $a*(x+y) = x*y$ form?

Question

If $a$ is a constant, what is the name of a curve of the form $a*(x+y) = x*y$? And how is this equation related to more this curve's more general equations/characteristics? Plotting this curve, I would risk calling it a hyperbola, but I'm not sure it is, or why it would be one. This equation is similiar to the equivalent parallel resistance formula for two resistors in eletronics (if rearranged: $a = \frac{x*y}{x+y}$).

Answer 1

$$xy=a(x+y)$$ $$xy-ax-ay=0$$ $$xy-ax-ay+a^2=a^2$$ $$(x-a)(y-a)=a^2$$ The curve is an hyperbola.

Answer 2

Every 2-degree equation in x and y represents a conic. So, if you see two branches in its graph, it is always going to be a hyperbola.