Two Foci Definition

Algebra 2 · Axiom Academy

LESSON The Two Foci Definition of an Ellipse Every point of an ellipse obeys one rule: its distances to two fixed points, the foci, always add to the same total, 2a . 1. A Constant Sum of Two Distances Fix two points, F_1 and F_2 , called the foci . An ellipse is every point P whose distances to the two foci add up to the same fixed total. Slide P around the curve and the two distances trade off — one grows exactly as much as the other shrinks — so their sum never moves. The defining property of every point on the ellipse Our running example, where the constant sum is 2a = 10 Two points on the ellipse reveal all of its measurements. At a vertex (the end of the long axis) the two distances lie flat along that axis, so their sum is exactly the full major-axis length 2a . At a co-vertex (the top of the short axis) the two distances are equal — each one is exactly a — and, with the center, they form a right triangle. At a vertex: the constant sum is the whole major axis At a co-vertex: the right triangle locates the foci The two fixed points, at on the major axis — here . a = 5 is half the long axis; b = 3 is half the short axis. , so each focus sits 4 units from the center. measures how stretched the ellipse is. With a = 5 , b = 3 , c = 4 , every point obeys d_1 + d_2 = 10 : at the vertex 9 + 1 = 10 , and at the co-vertex 5 + 5 = 10 .

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