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Is there a name for the curve in the plane defined by

$a/\|x - p\|^2 - b/\|x - q\|^2=\mathrm{constant}$

where $a$ and $b$ are fixed numbers and $p$ and $q$ are fixed points? How about if I don't square the denominators? How about if $a$ and $b$ are both $1$?

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I cannot offer names for your functions, but I was interested to see what they look like. Here is the function with the denominators unsquared, i.e., just the distances $||p-a||$ and $||q-b||$:
         Contour Plot
You might look at power Voronoi diagrams, which have a similar flavor (for multiple sites $p$, $q$).

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