Finding Foci

Algebra 2 · Axiom Academy

LESSON Finding Foci of Ellipses Locate an ellipse's two foci straight from its equation with c^2 = a^2 - b^2 — and see why that relationship has to hold. An ellipse is the set of points whose distances to two fixed foci add up to the same total. Watch a point P travel around the curve: the two focal distances each change, but their sum never does — it always equals 2a , the full length of the major axis. The defining property of every ellipse The foci sit on the major axis, each a distance c from the center Slide the point to a co-vertex — the end of the minor axis. By symmetry the two focal distances are equal there, and since they sum to 2a , each one equals a . That single fact builds a right triangle: the focal radius a is the hypotenuse, the semi-minor axis b is the vertical leg, and the center-to-focus distance c is the horizontal leg. When the larger denominator sits under x^2 , the major axis is horizontal and the foci lie on the x-axis at . Read off a^2 and b^2 , then compute c . Larger denominator under x^2 : foci at When the larger denominator sits under y^2 , the major axis is vertical, so the same method now places the foci on the y-axis at . The formula c^2 = a^2 - b^2 is unchanged — only the orientation flips. Larger denominator under y^2 : foci at You can now find any ellipse's foci from its equation — and you know why the formula works, not just how to use it.

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