Compound Locus
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A compound locus problem involves two, or possibly more, locus conditions occurring at the same time.  The different conditions in a compound locus problem are generally separated by the word "AND" or the words "AND ALSO".

Strategy for Solving Compound Locus Problems

If TWO locus conditions exist in a problem (a compound locus), prepare each condition separately ON THE SAME DIAGRAM.  After the two conditions are drawn separately, count the number of points where the two loci conditions intersect.
  (It may help to draw the locus locations as dotted lines.   Then find where the dotted lines cross.)

Steps:
1.  Draw a diagram showing the given information in the problem.  

2.  Read carefully to determine one of the needed conditions.  (Look for the possibility of the words "AND" or "AND ALSO" separating the conditions.)

3.  Plot the first locus condition.  If you do not see one of the locus theorems at work in the problem, locate one point that satisfies the needed condition and plot it on your diagram.  Then locate several additional points that satisfy the condition and plot them as well.  Plot enough points so that a pattern (a shape) is starting to appear, or until you remember the needed locus theorem for the problem. 

4.  Through these plotted points draw a dotted line to indicate the locus (or path) of the points.

5.  Repeat steps 2-4 for the second locus condition.

6.  Where the dotted lines intersect will be the points which satisfy both conditions.  These points of intersection will be the answer to the compound locus problem.

 

Consider:  A treasure is buried in your backyard.  The picture below shows your backyard which contains a stump, a teepee, and a tree.  The teepee is 8 feet from the stump and 18 feet from the tree.  The treasure is equidistant from the teepee and the tree AND ALSO 6 feet from the stump.  Locate all possible points of the buried treasure.

 

 

 

Description:

The red line represents the locus which is equidistant from the teepee and the tree (the perpendicular bisector of the segment).  The blue circle represents the locus which is 6 feet from the stump.  These two loci intersect in two locations.  The treasure could be buried at either "X" location.  Start digging!!

Note:  The red line and blue circle could have been drawn "dotted" to indicate that they are locus locations.