Wentworth's Plane and Solid Geometry · Book IX, §§ 945–959 (Prop. XXV) · Book 10: Book IX: Conic Sections
144. The hyperbola
Goal: Your student can define a hyperbola, draw one by points and with a bar and string, and name its foci, vertices, axes and eccentricity.
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 definition of the hyperbola (§§ 945–950), Proposition XXV (§ 951), and the corollaries and definitions (§§ 952–959).
The lesson, step by step
Warm up
Ask: Define an ellipse. What would happen if we used the difference of the distances instead of the sum?
The new idea
An hyperbola is a curve which is the locus of a point that moves in a plane so that the difference of its distances from two fixed points in the plane is constant.
§ 957: c² = a² + b². Notice the change from the ellipse.
Draw the figure
Do Prop. XXV: mark F and F' 8 cm apart and A, A' 6 cm apart, centred between them. For points X beyond F, swing arcs of radii A'X and AX from F' and F. Plot several points on each branch.
Walk through the proof
XXV: A and A' are points of the curve, for AF' − AF = AA' = 2a. For any X, the arcs meet at a point whose focal radii are A'X and AX, which differ by AA' = 2a.
§ 952: no point of the perpendicular at O can be on the curve, since it is equally far from both foci.
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.
Exercises
If you have a stick and string, try the method of § 950.
Wrap up
Ask: How many branches does a hyperbola have? (Two.)