In a 2018 blog post, I presented George Gamow’s pirate treasure problem, which can neatly be solved by capitalizing on the geometry of complex numbers. There’s more treasure to be had, however, so get ready for another adventure!

An island contains a giant boulder, a lighthouse, a cave, and a jail. Among a pirate’s belongings, you find the following note:

My treasure is five miles farther away from the giant boulder than it is from the lighthouse. It’s also six miles farther away from the cave than it is from the jail.

The Web Sketchpad model below (and here) shows the landmarks on the island. It also comes with several tools, including one that constructs hyperbolas given their foci and a point on the hyperbola. Can you use these tools to locate the treasure? (Note that the model substitutes centimeters for miles.)

It will help to recall the geometric definition of a hyperbola: The set of points P such that the difference of the distances from P to two fixed points (the foci) is constant.

The video at the end of this post demonstrates how the tools work. And if you’re interested in more interactive conic section materials, you’ll find my prior posts here.