Wentworth's Plane and Solid Geometry · Book IX, Exs. 885–910 · Book 10: Book IX: Conic Sections
143. Exercises on the ellipse
Goal: Your student proves and constructs several properties of the 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)
What's in the book
Pages 443–444: Exercises on the ellipse, Ex. 885–910.
The lesson, step by step
Warm up
Ask: State three facts about the tangent of an ellipse.
Theorems
Ex. 885. The major axis is the longest chord that can be drawn in an ellipse.
Ex. 892. The minor axis is the shortest diameter of an ellipse.
Ex. 896. The area of an ellipse is to the area of its auxiliary circle as the minor axis is to the major axis.
Constructions
Ex. 887. To draw a tangent and a normal at a given point of an ellipse.
Ex. 908. Given an ellipse, to find by construction the centre, the foci, and the axes.
A trammel
Ex. 903. If a straight line moves so that its extremities are always in contact with two fixed straight lines perpendicular to each other, then any point of the moving line describes an ellipse.
Try it: slide a ruler with its ends on two perpendicular edges and mark a point on it as it moves.
Wrap up
Your student explains one solution aloud.
Answers to the book's problems
Ex. 885: every point of the ellipse lies within its auxiliary circle (Prop. XIV), and no chord of that circle is longer than its diameter, the major axis. Ex. 895: b² = a² − c² = (a − c)(a + c). Ex. 896: πab : πa² = b : a = 2b : 2a. Ex. 908: two parallel chords have their middle points on a diameter; its middle point is the centre; a circle about the centre meets the ellipse in four points, giving the axes by symmetry; the foci are a from the ends of the minor axis.
Worked out for this site from the scan. If a number in your copy differs, trust the book.