Wentworth's Plane and Solid Geometry · Book IX, Props. XXXIII–XXXIV · Book 10: Book IX: Conic Sections

147. Asymptotes; the hyperbola as a section of a cone

Goal: Your student can define the asymptotes and prove they never meet the curve, and explain why a plane cutting both nappes of a cone gives a hyperbola.

⏱ About 40 minutes · 6 steps

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You'll need

  • The book, open to these pages
  • A ruler (straightedge) and a pair of compasses
  • Sharp pencil, eraser and plain paper (or a geometry notebook)
  • Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)

What's in the book

The asymptotes (§ 980), Propositions XXXIII–XXXIV (§§ 981–987).

The lesson, step by step

  1. Warm up

    Ask: What is the conjugate axis 2b of a hyperbola?

  2. The new idea

    The asymptotes of an hyperbola never meet the curve, however far produced.

    The section of a right circular cone made by a plane that cuts both nappes of the cone is an hyperbola.

  3. Draw the figure

    Draw a hyperbola, then the rectangle with sides 2a and 2b touching the curve at A and A'. Extend its diagonals: the curve gets closer and closer to them but never reaches them.

    Hold a lamp with a round shade near a wall: the light on the wall is bounded by hyperbolas.

  4. Walk through the proof

    XXXIII: if the asymptote touched the curve, the point would be where F'G meets it (§ 978); but F'G is parallel to the asymptote (§ 189), so they never meet.

    XXXIV: two spheres, one in each nappe, touch the plane at F and F'. From P, PF = PK and PF' = PK'; the difference is KK', the same for every point.

  5. Your student proves it back

    Close the book. Your student draws the curve and the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason. Point to each line as it is named.

    If a step is missing, ask "How do we know that?" rather than supplying it.

  6. Wrap up

    Ask: Name the three conic sections and say how the cutting plane lies for each. (Parallel to one element: parabola; cutting all elements of one nappe: ellipse; cutting both nappes: hyperbola.)

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