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I'm trying to find the parametric equation for $zx + zy - xy = 0$ or equivalent $z = \frac{xy}{x+y}$ but couldn't find any hint in the web neither in a couple of calculus books for this particular equation.

I have no clue about how to proceed :(

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Big hint: When $xyz\neq 0$, $$\frac{1}{z} = \frac{1}{y}+\frac{1}{x}$$

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