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Topos Uranos
English

Conic sections: characterization and graphs of the three conics

Summary

This lesson brings together in a single theory the curves studied in the earlier lessons. It presents the conic sections as the curves that result from cutting a double cone with a plane, recalls their canonical forms and shows that every second-degree equation without a term in xyxy represents a parabola, an ellipse, a hyperbola or one of their degenerate cases: a point, one or two lines, or no point at all. Completing the squares is enough to decide which curve it is and to read off its center or its vertex, its semi-axes and its foci. It ends with the eccentricity, which brings the three curves together in a single definition by means of a focus and a directrix, and with the intersection of conics and lines.

Learning objectives

  1. Recognize the conic sections as cuts of a double cone by a plane, including their degenerate cases.
  2. Bring a second-degree equation without a term in xyxy to its canonical form by completing squares, and decide from the signs of its coefficients which curve it represents.
  3. Obtain the center or the vertex, the semi-axes, the foci, the directrix and the asymptotes of a conic given by its general equation, and draw its graph.
  4. Use the eccentricity to describe the conics as loci, and the discriminant to study their intersections with lines.

The conic sections

Consider a double cone: the surface generated by a line, the generator, as it rotates about another line, the axis, which it meets at a fixed point, the vertex of the cone. The surface has two nappes, one on each side of the vertex. If it is cut by a plane that does not pass through the vertex, the resulting curve depends on the angle that the plane makes with the axis, compared with the angle that each generator makes with the axis.

  • If the plane is perpendicular to the axis, the curve is a circle.
  • If the plane makes with the axis an angle greater than that of the generator, it cuts all the generators of a single nappe and the curve is an ellipse, which is closed.
  • If the plane makes with the axis the same angle as the generator, that is, if it is parallel to a generator, it cuts a single nappe and the curve is a parabola, which is open.
  • If the plane makes with the axis an angle smaller than that of the generator, it cuts both nappes and the curve is a hyperbola, with two branches.

If the plane passes through the vertex of the cone, the intersection reduces to a point, to a line (a generator) or to two lines that meet at the vertex. These are the degenerate cases, and they will reappear later as solutions of certain second-degree equations, together with two others that are not obtained by cutting a cone: the pair of parallel lines, which results from cutting a cylinder, and the empty set. Note, however, that the circle is not a fourth family, but the ellipse that corresponds to the perpendicular position of the plane.

That the curves obtained by cutting the cone coincide with those that the earlier lessons defined by means of foci is Dandelin’s theorem. Its proof lies beyond the scope of this course, because it requires solid geometry (the spheres inscribed in the cone and tangent to the cutting plane, whose points of contact with the plane are the foci); that is why the cuts of the cone serve here only as a picture, and the definitions used are those based on distances.