In the interactive model below, point O represents the origin of a coordinate system. A circle sits with its center on the y-axes. Points B, C, and D mark where the circle intersects the axes. Segments BE and CF are parallel to the y-axis. Segment EF is parallel to the x-axis.
Drag the center of the circle along the y-axis and notice that point A stays in place. What curve do points E and F seem to trace?
Drag point A to a new location. Now clear your trace by clicking the red 'X' in the bottom right corner. Drag the center point again. How has your curve changed?
Explain why BO x OC = AO x OD.
Prove algebraically that points E and F trace a parabola. Hint: assume the coordinates of point F are (x, y). Rewrite the equality from the previous question in terms of x and y.