Asymptotes of a Hyperbola
Definition. The asymptote of a curve is a straight line approached by the curve line at infinity.
An oblique asymptote is determined by an equation of the form an oblique
y=kx+b.
To find the slope, we suppose that b=0; then
; .
We find b from the equation .
Passing to the limit, we obtain
Thus, the hyperbola has two asymptotes, passing through the origin:
; .
First, we construct a rectangle with sides 2à along the õaxis and 2b along the óaxis. We draw the diagonals in this rectangle and extend them; they are the asymptotes of the hyperbola.
y
2a 2b
F_{1} Ì_{1}(–à;0) 0 Ì_{2}(à;0) F_{2} x
From the vertices M_{1}(–a;0) and M_{2}(a;0), we draw the branches of the hyperbola approaching the asymptotes.
The directrices of a hyperbola. By definition, . By analogy with an ellipse, we find the difference of squares
,
.
Date: 20150102; view: 1203
