Equations of Circles
Last week, I was trying to help one of my students remember the general equation for circles. We were about done with the session, and we didn't have big plans afterwards, respectively, so I wanted to have some fun doing this. I asked him about the Pythagorean Theorem, and after a brief review of right triangles, he knew that the square of the length of the hypotenuse of a right triangle was equal to the sum of the squares of the lengths of the legs of that same triangle. Then, I drew a coordinate plane and an arbitrary right triangle with one leg on the x-axis and mentioned the Pythagorean again with the idiosyncrasies of this case. Next, I drew an arbitrary line segment in the first quadrant (for simplicity) with end points A and B, and with some hints, he derived the distance formula for points on a 2-dimensional coordinate plane. I had hoped he would have remembered it, but at least he knows how to derive it now. I followed with a question: what relationship exists between the set of all points of a circle and its center? He answered correctly that each and every point on the circle was at a fixed distance (r) away from the center that is equal to the radius of the circle. We finished by representing that statement as a special case of the distance formula where the center of the circle was represented by (h,k) and any point on the circle itself was of the form (x,y). We got rid of the square root by raising each side to the second power. It was a nice little exercise.