Wentworth's Plane and Solid Geometry · Book IX, Props. XXX–XXXII · Book 10: Book IX: Conic Sections
146. Tangents of the hyperbola
Goal: Your student can tell the concave from the convex side, construct the tangent at a point, and draw tangents from a point.
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 XXX–XXXII (§§ 969–979).
The lesson, step by step
Warm up
Ask: How is the tangent to an ellipse drawn at a point?
The new idea
The difference of the distances of any point from the foci of an hyperbola is greater than or less than 2a, according as the point is on the concave or convex side of the curve.
The tangent to an hyperbola at any point bisects the angle between the focal radii drawn to that point.
To draw a tangent to an hyperbola from a given point P on the convex side of the hyperbola.
Draw the figure
On your hyperbola, choose a point P, draw both focal radii, and bisect the angle between them: that is the tangent.
Walk through the proof
XXX: compare with the point where the line to F' crosses the curve (§ 138).
XXXI: take PG = PF on PF'; for any other point Q of the bisector, QF' − QF < GF' = 2a, so Q is on the convex side.
XXXII: as for the ellipse, using circles of radius PF about P and 2a about F'.
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: The ellipse tangent bisects the angle between one focal radius and the other produced; what does the hyperbola tangent bisect? (The angle between the focal radii themselves.)