Wentworth's Plane and Solid Geometry · Book IX, Props. XXVI–XXIX · Book 10: Book IX: Conic Sections
145. Symmetry and ordinates of the hyperbola
Goal: Your student can prove the symmetry of the hyperbola, find its focal radii, and relate its ordinates to the auxiliary circle.
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 XXVI–XXIX (§§ 960–968).
The lesson, step by step
Warm up
Ask: For the hyperbola, how are a, b and c related?
The new idea
An hyperbola is symmetrical with respect to its conjugate axis.
Any ordinate of an hyperbola is to the tangent from its foot to the auxiliary circle as b is to a.
Prop. XXVII: r = ed − a and r' = ed + a. Prop. XXIX: the square of the ordinate is to the product of the distances from its foot to the vertices as b² : a².
Draw the figure
Draw a hyperbola with its auxiliary circle (diameter AA'). From the foot M of an ordinate PM, draw the tangent MQ to the circle.
Walk through the proof
XXVI: fold on the conjugate axis, as for the ellipse.
XXVII: r'² − r² = 4cd and r' − r = 2a, so r' + r = 2ed.
XXVIII: PM² works out to (b²/a²)(d² − a²), and MQ² = d² − a².
XXIX: MQ² = AM × A'M (a tangent and a secant from M).
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: Compare Props. XIII and XXVII. How are the focal-radius formulas of the ellipse and hyperbola alike and different?