Wentworth's Plane and Solid Geometry · Book IX, Props. XVI–XIX · Book 10: Book IX: Conic Sections
140. Inside and outside; tangents of the ellipse
Goal: Your student can relate the ordinate to the segments of the major axis, tell inside from outside, and construct tangents to an ellipse.
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 XVI–XIX (§§ 919–930).
The lesson, step by step
Warm up
Ask: Construct the tangent to a parabola at a point. What do you expect for an ellipse?
The new idea
The sum of the distances of any point from the foci of an ellipse is greater than or less than 2a, according as the point is without or within the curve.
The tangent to an ellipse at any point bisects the angle between one focal radius and the other produced.
To draw a tangent to an ellipse from an external point.
Prop. XVI: the square of the ordinate is to the product of the segments of the major axis as b² : a². § 920: the latus rectum is the third proportional to the major and minor axes.
Draw the figure
On your string ellipse, pick a point P, draw both focal radii, extend one past P, and bisect the angle between it and the other: that is the tangent.
Walk through the proof
XVI: from Prop. XIV and QM² = AM × A'M.
XVII: compare with a point P of the curve between the lines to the foci (§ 100).
XVIII: on FP produced take PG = PF'. For any other point Q of the bisector, QG = QF', so QF + QF' = QF + QG > FG = 2a: Q is outside.
XIX: find G on the circle of radius 2a about F' with PG = PF; F'G meets the curve at the point of contact.
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: A whisper at one focus of an elliptical room is heard clearly at the other focus. Why? (§ 935: the normal bisects the angle between the focal radii, so sound from one focus reflects to the other.)