The diagram below shows a red unit circle with a dynamic blue circle intersecting it at (0, -1) and (0, 1). The measures of the x-coordinates of the dynamic blue circle intersecting the x-axis is shown, as well as the product of these two x-coordinates.
1) Drag the point O_{B} and observe what happens regarding the product of the x-coordinates.
2) What do you notice about the path of H, the highest point of the blue circle? Can you find a formula for the curve?
3) Can you prove your observations?