- #1
flyingpig
- 2,579
- 1
Homework Statement
I`ll try to make this as orderly as possible, but I've got so many questions about it
1. The most "general" form of a hyperbola are
[tex]\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1[/tex]
[tex] \frac{y^2}{b^2}- \frac{x^2}{a^2}= 1[/tex]
Now my question is, the first one opens with the x-axis, the second one opens with the y-axis. My question is, I am never going to be able to rememeber them, even if i draw out my asympotetes I am not going ot be able to deduct with which axis does the hyperbola open.
Also just another side question, the asympotetes are negatives of each other, but when I graphed it, they are also perpendicular to each other. Now here is the thing, how come they aren't negative reciprocal of each other?
2. Sometimes we call [tex]xy = 1[/tex] as a hyperbola how do I convert those from (1) to this form?
3. How do the hyperbolic functions apply to (2)?
4. I never understood this, that's say [tex]\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1[/tex]
Now I am going to rewrite it as [tex]y = \pm \frac{b}{a}\sqrt{x^2 - a^2}[/tex]
Now I just want to look at [tex]\sqrt{x^2 - a^2}[/tex]
How do I recognize that [tex]\sqrt{x^2 - a^2}[/tex] will give me a curve and not a straight line? I used to think that the square and the square root "cancels" and the a doesnt' matter. I was wrong.