Geometric locus of $M$ such that $|MA|^2 |MB|^2=a^2$ I want to find the geometric locus of point $M$ such that $|MA|^2 |MB|^2=a^2$ where $|AB|=2a$, Solving algebraic equation is not hard but I can't figure out the shape of this curve. Can anybody help?
 A: If you want to toy with these shapes, you can try this:


*

*Download and install Cinderella (free version should be enough)

*Start it and add four free points ($A$ through $D$) to the construction

*Press Ctrl+Enter to open a command input box

*Check the “Permanent” checkbox next to that

*Paste the following code:
colorplot(if(dist(#,A)^2*dist(#,B)^2<(dist(A,B)/2)^2,[1,1,0],[0,0,1]),C,D,pxlres->1,startres->8);


*Press enter
You should see a blue plot region, its shape determined by the points $C$ and $D$, and within this one or two yellow blobs, centered around $A$ and $B$ and delimited by the curve you asked about. Something like this:

The code in detail:
colorplot(    // plot pixels depending on function value
  if(         // function is a case distinction
    dist(#,A)^2 * dist(#,B)^2 < (dist(A,B)/2)^2, // sign for your curve
    [1,1,0],  // yellow inside
    [0,0,1]   // blue outside
   ),
 C,D,         // these points control the plotting area
 pxlres->1,   // draw at finest resolution eventually
 startres->8  // draw at coarse resolution first for smooth movements
);

Also see the CindyScript reference of the colorplot function.
A: If points $A,B$ are fixed and negative distances are ruled out, the loci are quartic curves, Ovals of Cassini.  (Incorrectly ) he assumed them at first to be planetary orbit loci.
Ovals_Cassini1
Ovals_Cassini2
A: Use GeoGebra's LocusEquation command to create an implicit locus curve. Then you can drag the free points and check how the curve is dynamically changing.
