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Topos Uranos
English

The equation of the ellipse and of the circle

Summary

This lesson defines the ellipse as the set of points of the plane whose sum of distances to two fixed points, the foci, is constant, and derives its reduced equation from that definition, checking that the process neither adds nor loses points. It identifies the semi-axes and the focal distance, and their Pythagorean relation; it translates the equation to obtain ellipses with their center at any point and a horizontal or vertical major axis, and obtains the circle as the case in which the two foci coincide. It ends with the expanded equation, from which, by completing squares, one decides whether it describes an ellipse, a circle, a single point or no point at all.

Learning objectives

  1. State the definition of the ellipse by means of its foci, justify the condition that the sum of the distances must satisfy and derive the reduced equation from it.
  2. Recognize the center, the semi-axes, the vertices and the foci of an ellipse with axes parallel to the coordinate axes.
  3. Identify ellipses and circles in an expanded equation by completing squares, and recognize the cases in which the equation describes a point or no point at all.
  4. Solve problems of intersection, of tangency and of loci that lead to ellipses.

Definition of the ellipse

As in the lesson on the equation of parabolas, the starting point is a geometric property expressed by means of distances, which are computed with the formula seen in the lesson on the equation of a line.

Let F1F_1 and F2F_2 be two points of the plane and aa a number greater than half the distance between them. The ellipse with foci F1F_1 and F2F_2 is the set of the points PP of the plane whose sum of distances to the foci is equal to 2a2a.

The condition imposed on aa is not a minor detail. By the triangle inequality, proved together with its equality case in the lesson on vectors in the plane, the sum of the distances from any point PP to the foci is greater than or equal to the distance between them, and equality holds only at the points of the line segment joining the foci. It follows that, if 2a2a were equal to the distance between the foci, the set would be that line segment, and if it were smaller, it would contain no point; only when 2a2a exceeds the distance between the foci is a curve obtained.

The midpoint of the foci is the center of the ellipse, and the line through both of them, its focal axis. If 2c2c is the distance between the foci, the number cc is the focal distance, and the condition above is written a>c≥0a > c \geq 0.