   Chapter 3.5, Problem 57E

Chapter
Section
Textbook Problem

# Show that the lines y = ( b / a ) x and y = − ( b / a ) x are slant asymptotes of the hyperbola x 2 a 2 − y 2 b 2 = 1

To determine

To show:

That the lines of y=bax and y=-bax are slant asymptotes of the hyperbola

Explanation

1) Concept:

Slant asymptote: the line y=mx+b is called a slant asymptote if

limx±fx-(mx+b)=0

2) Given:

x2a2- y2b2=1

3) Calculation:

Solve the hyperbolic equation for y

x2a2- y2b2=1

y2=b2x2a2-1

Factoring out 1a2

Therefore,

y2=b2a2x2-a2

Taking Square root of both sides,

y=±ba x2-a2

Use the concept of slant asymptote to check whether bax is slant asymptote of this function.

limx± ∞ba x2-a2-bax

Factoring out ba

=balimx± ∞ x2-a2-x

Multiply this limit by  x2-a2+ x x2-a2+x

Therefore,

=balimx± ∞ x2-a2-x · x2-a2+ x x2-a2+x

=balimx± ∞x2-a2-x2 x2-a2+x

=balimx± ∞-a2 x2-a2+x=0

Therefore, bax is slant asymptote of this function

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