Prove equations for asymptotes of standard hyperbola

• JsStewartFan
In summary, the asymptotes of a standard hyperbola are y=-(b/a)x and y= (b/a)x. The equation for the asymptotes is y=mx+b.
JsStewartFan

Homework Statement

Prove the equation(s) for the asymptotes of a standard hyperbola. That is, prove that the asymptotes for the hyperbola x^2/a^2 - y^2/b^2 = 1 are

y = -(b/a)x and
y = (b/a)x

where foci are at (c,0) and (-c,0); vertices are at (a,0) and (-a,0); difference in distances between the two foci and any point on the hyperbola is a constant 2a or -2a; and

c^2 = a^2 + b^2.

Homework Equations

If lim(x→∞) [f(x)- (mx+b) ]= 0, then y= mx + b is the equation of an asymptote of f(x).

The Attempt at a Solution

Using the definition of the asymptote in terms of the limit difference in (2) above, I focused on the first quadrant of the hyperbola and came up with this sequence of equations, but since (-b) does not equal zero, I have an error somewhere:

lim(x→∞)[b/a (x-a)+ bx/a]
= lim(x→∞) [(bx-ba)/a + bx/a]
=lim(x→∞) [(bx-ba+bx)/a]
=lim(x→∞) [(-ba)/a]
= (-b)lim(x→∞)1
= (-b)• 1
= -b ≠ 0

By the way, this is in James Stewart's Calculus - Early Transcendentals, 5th edition, section 4.5, problem 67, page 324. There's no answer in the back of the book. I am reviewing hyperbolas for a high school teacher certification test and got off on this "tangent" because I like to derive or prove equations instead of just memorizing them (when I have time).

Do you have to use the definition in terms of the limit difference?

By the limit difference, I just meant the definition for the asymptotes that says if the limit as x approaches infinity of (f(x) - (mx + b)) equals zero, then that y=mx+b is an asymptote of the function f(x), which is the hyperbola in question. It's listed under related equations.

I wasn't sure what you meant by liking my whiteboard, but I used MS Word 2007 with Equation Tools to write out the limit expression. Then I copied and pasted. I've never used Latex Reference before, so I used what I knew.

Also, could you help me with navigation? I can't figure out a quick way to get to the homework and course work help section.

Thanks!
Terry

Oh, I read your question wrong. I think since Mr. Stewart only explained slant asymptotes in terms of the limit definition, that he's expecting me to use it here. This is a problem right after that section. I know I can use setting the hyperbola to zero instead of 1 (saw that in another post here), but I should be able to use limits to prove it, too.

Thanks,
Terry

I finally found the homework and course work section under Science, so I'm OK there.
Thanks,
Terry

QuarkCharmer:
I think I've proved this two ways - one I found in Sullivan's Algebra & Trigonometry book (proved in the text), ed. 7, page 793. The other one I proved using the difference of limits as I wanted to, but I had to separate the proof into the four quadrants, since the hyperbola is not a function unless I do that. It turned out OK, I think.

I'd like to send it to you, but I took about 30 minutes after I copied and pasted my new Word document, trying to fix all the spaciing and so forth - then I was logged out automatically and lost it all. Let me know if there's some way I can get the Word document to you.

I'd appreciate you looking at it for me, and maybe you can post it to this question if correct.

Thanks,
Terry

1. What is an asymptote?

An asymptote is a line that a curve approaches but never touches. In other words, as the curve extends to infinity, it gets closer and closer to the asymptote but never actually intersects it.

2. How do you find the equations for the asymptotes of a standard hyperbola?

To find the equations for the asymptotes of a standard hyperbola, use the formula y = ± (b/a)x, where a is the distance from the center of the hyperbola to the vertices and b is the distance from the center to the co-vertices.

3. What are the properties of the asymptotes of a hyperbola?

The asymptotes of a hyperbola have the following properties:

• They are straight lines that intersect at the center of the hyperbola.
• Their slopes are equal to ± (b/a), where a and b are the lengths of the semi-major and semi-minor axes of the hyperbola, respectively.
• They are perpendicular to each other.
• The hyperbola approaches but never touches the asymptotes.

4. Can a hyperbola have more than two asymptotes?

No, a hyperbola can only have two asymptotes. This is a defining characteristic of a hyperbola.

5. How do you graph the asymptotes of a standard hyperbola?

To graph the asymptotes of a standard hyperbola, you can plot the vertices and co-vertices of the hyperbola, then draw two lines through the center of the hyperbola that intersect at the vertices. These lines will be the asymptotes of the hyperbola.

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