If we know an ellipse is centered at the origin, we know that there will be only x^2 and y^2 terms (rather than squared binomials.) The foci always lie along the major axis, so the placement of the foci will tell us whether the ellipse is "surfboard" style, meaning major axis vertical and x^2 over b^2, or "hamburger" style, meaning major axis horizonal and x^2 over the a^2 term. The focus distance from the center is "c," and the co-vertex distance is "b." Using this information and the relationship a^2 - b^2 = c^2, we can put it all together to write the equation of the ellipse.
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