Is there a general form for the hyperbola $\frac1x+\frac1y=-\frac2{37}$?

Question

I know that $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ or $xy=c^2$ or $x^2-y^2=a^2$ represents hyperbola.

But I came across the following equations that according to desmos and wolfram are hyperbola. I wonder what their general form is.

• $\frac1x+\frac1y=-\frac2{37}$ (wolfram)
• $y=\frac{x-1}{x+1}$ (wolfram)

I came across the first equation in a Physics problem and second one on toppr website while I was looking up the first one.

Answer 1

I was multiplying by the denominators but wasn't reaching anywhere.

@Intelligenti pauca's comment made me reverse engineer the equation, and now I can see how the given equation is a rectangular hyperbola.

$$\frac{y+x}{xy}=-\frac1{18.5}\\\implies18.5x+18.5y+xy=0$$

Adding $(18.5)^2$ on both sides, we get

$$(x+18.5)(y+18.5)=(18.5)^2$$