Wentworth's Plane and Solid Geometry · Book IX, §§ 850–861 (Prop. I) · Book 10: Book IX: Conic Sections
133. The parabola
Goal: Your student can define a parabola by its focus and directrix, draw one by points and with a string, and name its vertex, axis, ordinate, abscissa and latus rectum.
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)
- A ruler, a draftsman's triangle (optional), string and a pin
What's in the book
The definition of the parabola (§§ 850–852), Proposition I (§ 853), definitions and corollaries (§§ 854–861).
The lesson, step by step
Warm up
Ask: What is the locus of points equally distant from two points? from a point and ...? Today: a point and a line.
The new idea
A parabola is a curve which is the locus of a point that moves in a plane so that its distance from a fixed point in the plane is always equal to its distance from a fixed line in the plane.
To construct a parabola by points, having given its focus and its directrix.
§ 861: the latus rectum is equal to 4AF.
Draw the figure
Draw a line (the directrix) and a point F 4 cm away (the focus). Do Prop. I: through points M on the axis draw lines parallel to the directrix, and from F swing arcs of radius DM to cut them. Find ten points and join them smoothly.
If you have a draftsman's triangle, try the string method of § 852.
Walk through the proof
I: P lies on a line parallel to the directrix at distance DM from it, and FP = DM by construction; so P is as far from the focus as from the directrix.
§ 858: the curve is symmetrical about its axis. § 860: it is not a closed curve. § 861: at L, LF = LG = DF = 2AF, so the latus rectum is 4AF.
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.
Wrap up
Ask: Where do you see parabolas? (The path of a thrown ball; the shape of a flashlight reflector.)
Tip
Book IX is a short introduction to the conic sections and can be taken more lightly. The drawing is the fun part: let your student draw each curve carefully and large.