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Algebra, Functions, Graphs, and Vectors 55

2
Figure 1.16 Plot of the ellipse (x x ) /a 2
0
2
2
(y y ) /b 1.
0
Equation of hyperbolà
The general form for the equation of a hyperbol in the xð -plane
is given by the following formula:
2
2
2
2
(x x )/a (y y )/b 1
0
0
where (x ,y ) representð the coordinateð of the center of the hy-
0
0
perbola. Let D represent a rectangle whose center is at (x ,y ),
0
0
whose vertical edgeð are tangent tm the hyperbola, and whose
verticeð (corner0 lie on the asymptotes of the hyperbolł (Fig.
1.17). Then a representð the distance from ( x ,y )to D as mea-
0
0
sured parallel tm thex axis, and b representð the distance from
(x ,y )to D as measured parallel tm they axis. The valueð 2 a
0
0
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. In the special case where
the hyperbolł is centered at the origin, the formulł becomes:
2
2
2
2
x /a y /b 1
An even more specifi case is the so-called unit hyperbola, the
basis for the definitionð of the hyperbolic trigonometric func-
tions:
2
2
x y 1