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

134. Ordinates and tangents of the parabola

Goal: Your student can prove that the ordinate is the mean proportional between the latus rectum and the abscissa, tell inside from outside, and construct the tangent at a point.

⏱ About 40 minutes · 7 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 II–IV (§§ 862–879) with corollaries, and the definitions of tangent, normal, subtangent and subnormal.

The lesson, step by step

  1. Warm up

    Ask: Define the parabola, its focus, directrix, vertex and latus rectum.

  2. The new idea

    The ordinate of any point of a parabola is the mean proportional between the latus rectum and the abscissa.

    Every point within the parabola is nearer to the focus than to the directrix; and every point without the parabola is farther from the focus than from the directrix.

    Prop. IV and § 869: the bisector of the angle between PF and the perpendicular from P to the directrix is tangent to the curve at P.

    § 876: the subtangent is bisected by the vertex. § 877: the subnormal is equal to half the latus rectum.

  3. Draw the figure

    On your parabola from last time, pick a point P. Draw PF and PC (perpendicular to the directrix), bisect the angle FPC, and check that the bisector just touches the curve.

  4. Walk through the proof

    II: PM² = PF² − FM² = DM² − FM² = (DM − FM)(DM + FM) = 2AF × 2AM.

    III: for Q inside, QF < QP + PF = QP + PC = QC.

    IV: if X is any other point of PT, XF = XC > XE (the perpendicular), so X is outside (§ 866).

  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. Exercises

    If AF = 1 (latus rectum 4), find the ordinate at abscissa 4 (√16 = 4) and at abscissa 9 (6). Plot these points on your drawing.

  7. Wrap up

    Ask: Light from the focus hits the parabola. Where does it go? (§ 878: it is reflected parallel to the axis. That is how a flashlight works.)

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