Where is the right triangle in sides a, b, c, for a hyperbola?

[symbolipoint]

An ellipse has some model standard form values, a, b, and c which are easily enough to identify from the graph and parts of the graph related to the ellipse's graph. Seeing the right triangle relating a, b, and c, is easy enough. The Pythagorean Theorem is used to relate these three values.


I have been hoping to find and SEE the values of a, b, and c in their right triangle for a hyperbola, but I just cannot find this triangle. I have looked in a couple of College Algebra textbooks, and the most that is done is to assign a substitution as part of the derivation of the equation for the hyperbola. That just does not help to show where to find the right triangle. $$a^2+b^2=c^2$$ comes simply from the part of $$c^2-a^2$$
but no picture to support it.

 

[SteamKing]

An ellipse has some model standard form values, a, b, and c which are easily enough to identify from the graph and parts of the graph related to the ellipse's graph. Seeing the right triangle relating a, b, and c, is easy enough. The Pythagorean Theorem is used to relate these three values.


I have been hoping to find and SEE the values of a, b, and c in their right triangle for a hyperbola, but I just cannot find this triangle. I have looked in a couple of College Algebra textbooks, and the most that is done is to assign a substitution as part of the derivation of the equation for the hyperbola. That just does not help to show where to find the right triangle. $$a^2+b^2=c^2$$ comes simply from the part of $$c^2-a^2$$
but no picture to support it.

The right triangle is located between the center of the hyperbola, the vertex, and the point above the vertex lying on the asymptote.

See this article:

https://mysite.du.edu/~jcalvert/math/hyperb.htm

If we have a hyperbola whose center C is at the origin, the vertices V1 and V2 will be located at the points ($\pm$a, 0), while the asymptotes have the equations y = $\pm$(b/a)x. When x = a, at the positive vertex V1, y = b, and the relation

c$^{2}$ = a$^{2}$ + b$^{2}$ is also satisfied.

 

[symbolipoint]

Hi SteamKing,

I looked at and read some of the article. Nice, but a little difficult to follow. I still do not see/find a clear way to understand thinking from a and c, to the meaning and value of b. I can well accept that the b value and placement is realistic and meaningful; but I have trouble thinking through a good way to reach b from the hyperbola and a and c.

I have the right feeling that a rectangle must be associated with a hyperbola because this would correspond to what occurs in an ellipse. The ellipse has a, b, an c, easy to see. Then, one could expect to find somewhere, a meaningful rectangle(in a hyperbola). The rotation (for the hyperbola) of the segment from center to either focus (the one on the right) then will meet the asymtpote, but then this seems strange to understand. I become lost at that. I have no way to think why to make this rotation. The asymptote and slope of the asymptote then seems to involve b, but then the b is what I am trying to understand. I could go on and just use b on faith and could answer some common academic questions about hyperbolas at the "Algebra 2" and "College Algebra" level, but I was just hoping to get inside the meaning and not need to use just faith. I feel like this lack of understanding about a hyperbola will be an obstacle to leaning conic sections and more about coordinate geometry and much more about Mathematics.

 

[SteamKing]

I don't know why you are dragging 'faith' into this situation.

As for an ellipse, if a is the length of the semi-major axis and b is the length of the semi-minor axis, then c is the distance between these two points. It's not clear why this is significant to you or to the construction of an arbitrary ellipse.

The equation of the asymptotes to the hyperbola are as stated. They are straight lines. I don't understand why this is difficult to grasp.

 

[symbolipoint]

I could not explain the difficulty any better. The ellipse can be drawn and a, b, and c are easily seen. Applying b to the standard form equation then becomes fairly easy. If I would understand the hyperbola, I would need to start with ONLY a and c, and look for a a way to understand b without any reference to b; only to what can be found with the graph , and a and c. Why I use "faith" for this is because I need no such faith in understanding the ellipse. I remain stuck in understanding these for the hyperbola.