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

139. Symmetry, focal radii and the auxiliary circle

Goal: Your student can prove that the ellipse is symmetrical about its minor axis, find its focal radii, and construct it from its axes.

⏱ 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

Propositions XII–XV (§§ 911–918).

The lesson, step by step

  1. Warm up

    Ask: What is the relation between a, b and c?

  2. The new idea

    An ellipse is symmetrical with respect to its minor axis.

    The ordinates of two corresponding points in an ellipse and its auxiliary circle are in the ratio b : a.

    To construct an ellipse by points, having given its two axes.

    Prop. XIII (§ 915): the focal radii are r' = a + ed and r = a − ed, where d is the abscissa.

  3. Draw the figure

    Do Prop. XV: draw two circles with centre O, radii a = 6 cm and b = 4 cm. Draw lines from O; where each meets the big circle draw a line parallel to the minor axis, and where it meets the small circle a line parallel to the major axis. Their meeting points lie on the ellipse.

  4. Walk through the proof

    XII: fold the figure on the minor axis; F falls on F', so Q's focal radii equal P's.

    XIII: subtract the squares of the two right triangles: r'² − r² = 4cd, and r' + r = 2a.

    XIV: PM² = b²(a² − d²)/a² and QM² = a² − d².

    XV: the parallel lines cut OQ and OM proportionally, so PM : QM = b : a.

  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: If you squash a circle in one direction by the ratio b : a, what do you get? (An ellipse, Prop. XIV.)

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