Geometry: An Interactive Journey to Mastery

Folding and Conics

Lesson 29

Topics
x The parabola, ellipse, and hyperbola via folding.
x The parabola, ellipse, and hyperbola via conic sections.
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x ellipse: Given two points F and G in the plane, the set of all
points P for which the sum of distances FP + PG has the same
constant value traces a curve called an ellipse. The points F
and GDUHFDOOHGWKHIRFLRIWKHHOOLSVHͼ௘6HHFigure 29.1௘ͽ
x hyperbola: Given two points F and G in the plane, the set of all
points P for which the differences of distances FPíPG and
GPíPF have the same constant value traces a curve called a
hyperbola. The points F and G are called the foci of the hyperbola.
ͼ௘6HHFigure 29.2௘ͽ
x parabola: Given a line m and a point F not on
that line, the set of all points P equidistant from m
and P form a curve called a parabola. F is called
the focus of the parabola, and m is its directrix.
Summary
The ancient mathematical topic of conic sections makes
appearances in many realms, including astronomy, physics,
and medicine. In this supplemental lesson, we construct the
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and then by the classic approach of slicing a cone. We show that these two methods of construction do indeed
yield precisely the same set of curves.

P

FP

FP + PG = k

PG

FG

Figure 29.1

P

FP íPG k

GP íPF k

FG

Figure 29.2

F

m

Figure 29.3