(Probably) simple question: Asymptotes of a Hyperbola

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In summary, the conversation discusses finding asymptotes for a hyperbola and the process of setting the equation to 0 to find them. The individual asking the question initially arrives at an incorrect answer, but is reassured by the expert that their answer is correct and the book is wrong. The conversation also includes a side note about wanting the expert's whiteboard.
  • #1
sltungle
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Okay, before I start: I'm sorry for what is probably going to be an absurdly easy question and I'm probably going to seem like a complete moron to everyone here, however the way I see it the only way I'll learn is by asking questions (and that's just what I've did ever since I could speak)... so... help would be appreciated.

Find the asymptotes for the hyperbola: 2x^2/3 - y^2/2 = 1



Equations: Hyperbola: (x-h)^2/a^2 - (y-k)^2/b^2 = 1, Asymptote: ±b/a (x-h) + k



Attempt:

DSCF9170.jpg


I keep arriving at the answer ±2/√3 yet the answer in the back of the book is ±4/√3. What am I doing wrong?
 
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  • #2
Here's the easy way to think about "asymptotes". The whole point of asymptotes is that the hyperbola gets closer and closer to them as x and y get larger and larger. Now if x is extremely large, [itex]x^2[/itex] will be even larger and so will [itex]x^2/a^2[/itex] for constant a. Similarly, [itex]y^2/b^2[/itex] will be an extemely large number. Comparatively, the "1" on the right side of the equation is very small and negligible. If we drop it from the equation of the hyperbola, we get
[tex]\frac{x^2}{a^}- \frac{y^2}{b^2}= 0[/tex]
[tex]\left(\frac{x}{a}- \frac{y}{b}\right)\left(\frac{x}{a}+ \frac{y}{b}\right)= 0[/tex]

From that, x/a= y/b and x/a= -y/b are the two asymptotes.

Here, [itex]2x^2/3= x^2/(3/2)[/itex] so [itex]a= \sqrt{3/2}[/itex] and [itex]b= \sqrt{2}[/itex].

The asymptotes are given by [itex]x/(\sqrt{3/2}= y/\sqrt{2}[/itex] so that [itex]y= \sqrt{2}x/\sqrt{3/2}x= (2/\sqrt{3})x[/itex] and [itex]x/\sqrt{3/2}= -y/\sqrt{2}[/itex] so that [itex] y= -(2/\sqrt{3})x[/itex].

You are right and the book is wrong!
 
  • #3
Ahh. Thank you very much :) I was a bit worried I was missing something obvious. Glad to know I'm not.

Thanks again.
 
  • #4
Dear HallsofIvy,

Thanks for your easy-to-follow explanation of why you replace the 1 with a zero to find the asymptotes. I had gotten that tidbit from another website, but they didn't explain why you set the original equation to 0 instead of 1.
 
  • #5
As a side note, I want your whiteboard.
 

1. What is an asymptote?

An asymptote is a line that a curve approaches but never touches. It can be thought of as the "limit" of the curve.

2. How many asymptotes does a hyperbola have?

A hyperbola has two asymptotes, one for each branch of the curve.

3. How do you find the equations of the asymptotes of a hyperbola?

To find the equations of the asymptotes, you can use the formula y = mx + b, where m is the slope of the asymptote and b is the y-intercept. The slope can be found by taking the ratio of the coefficients of the first degree terms of the hyperbola's equation. The y-intercept can be found by setting the x-value to 0 and solving for y.

4. Do hyperbolas always have vertical or horizontal asymptotes?

No, hyperbolas can have both vertical and horizontal asymptotes, or just one or the other. It depends on the orientation of the hyperbola and the coefficients in its equation.

5. Can the asymptotes of a hyperbola intersect the curve?

No, by definition, asymptotes never intersect the curve. They can only approach it infinitely close.

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