I see that you got it. I'll answer it anyway, for the sake of completeness.Originally Posted bypidgezero_one

The general form for an ellipse is:

where the center of the ellipse (the geometric center, not one of the foci) is at the coordinates (h,k).

Since we know the center is at (0,0) we know that h = k = 0 and the ellipse equation is:

Now, we have two points on the ellipse: (2,-2) and (-2/3,10/3). Putting these points into the ellipse equation gives us (respectively)

1.

and

2.

So we have two equations for a and b. This should suffice to solve the system. Equation 1 gives us:

Putting this into equation 2 gives us:

Since a is the length of the semi-major or semi-minor axis, we choose the positive value.

This gives a value for b as: .

Thus the equation for the ellipse is:

.

-Dan