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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 2 along the -axis and 2b along the -axis. We draw the diagonals in this rectangle and extend them; they are the asymptotes of the hyperbola.

y

2a 2b

F1 1(;0) 0 2(;0) F2 x

 

 

From the vertices M1(a;0) and M2(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: 2015-01-02; view: 892


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