Deriving parametric equations of ellipse

In deriving parametric equations of an ellipse, we're not sure where to put the origin. But let's recall that the parametric equations for a circle of radius r are x = r cos t
y = r sin t
0 ¾ t ¾ 2¼ if the origin is at the center of the circle, and try to generalize. If we trace out the circle using these equations with any positive value of r (perhaps with the help of a graphing calculator) we see that the moving point progresses from the point (r , 0) counterclockwise to the point (0, r ), then on around to (–r , 0), then down to (0, –r ) and back to the start.

For the ellipse we'd like to do the same, except we want to go to points (a , 0) and (–a , 0) on the x -axis and to points (0, b ) and (0, –b ) on the y -axis.

But what determines how far out we go on each axis? The coefficient of cos t in the expression for x tells us how far out we go on the x -axis. What would happen if we changed that coefficient from r to a ? And changed the coefficient in the equation for y from r to b ? We'd have

x = a cos t
y = b sin t
0 ¾ t ¾ 2¼ If we trace this figure out for any positive values of a and b , we see that indeed we get the ellipse with horizontal axis 2a and vertical axis 2b , as desired.