Some Strange Circles
I was trying to construct a simple, introductory intersection problem for the first day of Calculus class, so I started with a well-behaved circle:
This is a circle centered at (3,2) with radius 5. I picked two points on the circle, (0,6) and (7,-1), found the equation of the line between them, and put together my system of equations:
So I had successfully reverse-engineered my circle-and-line intersection problem with two nice solutions: (0,6) and (7,-1).
Unfortunately, I made a typo on the handout. At the end of the left side of the circle equation I wrote ” + 12″ instead of ” – 12″.
So all my work was for naught. Or so I thought. Turns out, at least two amazing things happen:
First, the new circle still ends up having a nice radius, namely 1. What’s even more amazing is that the new circle also ends up having two nice intersections with the given line, (3,3) and (4,2)!
I wish my intentional work always turned out as well as this mistake!