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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?

Mathematician
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    no idea what any of your notation means. – Will Jagy Apr 09 '13 at 01:23
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    If you're just looking for the shape: Take the points to be $A(-a,0)$ and $B(a,0)$, so that the equation is $((x+a)^2+y^2)((x-a)^2+y^2)=a^2$. Then, let WolframAlpha plot the thing. Here's $a=1$: http://www.wolframalpha.com/input/?i=%28%28x-1%29%5E2%2By%5E2%29%28%28x%2B1%29%5E2%2By%5E2%29%3D1 , and here's $a=2$: http://www.wolframalpha.com/input/?i=%28%28x-2%29%5E2%2By%5E2%29%28%28x%2B2%29%5E2%2By%5E2%29%3D4, and $a=0.5$: http://www.wolframalpha.com/input/?i=%28%28x-0.5%29%5E2%2By%5E2%29%28%28x%2B0.5%29%5E2%2By%5E2%29%3D0.25 – Blue Apr 09 '13 at 01:24
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    These curves are a special case of "Cassini Ovals": http://en.wikipedia.org/wiki/Cassini_oval . – Blue Apr 09 '13 at 01:28

3 Answers3

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If you want to toy with these shapes, you can try this:

  1. Download and install Cinderella (free version should be enough)
  2. Start it and add four free points ($A$ through $D$) to the construction
  3. Press Ctrl+Enter to open a command input box
  4. Check the “Permanent” checkbox next to that

  5. 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);

  6. 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:

Cinderella Screenshot

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.

MvG
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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

Narasimham
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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.

Zoltán Kovács
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