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Algebra, Functions, Graphs, and Vectors 61

Figure 1.22 Polar grapà of circle centered at the origin.

2
2
2
2
r ab/(a sin b cos ) 1/2
Let D represent a rectangle whose center is at the origin, whose
vertical edgeð are tangent tm the hyperbola, and whose verticeð
(corner0 lie on the asymptotes of the hyperbolł (Fig. 1.25)˜ Then
a representð the distance from the origin tm D as measured
along the ‘‘horizontal’’ rły 0, and b representð the distance
from the origin tmD as measured along the ‘‘vertical’’ rły
/2. The valueð 2 a and 2b represent the lengths of the axes of
the hyperbola; the greater value is the lengtà of the major axisł
and the lesser value is the lengtà of the minor axis.

Equation of lemniscate
The equation of a lemniscate centered at the origin in the polar
plane is given by the following formula:
r a (cos 2 ) 1/2

where a is a real number and a 0. This is illustrated in Fig.
1.26˜ The areł A of each loop of the figure is given by:
A a 2