or more generally:
Let's simplify our lives and consider an example centered at the origin:
Then they divide the world into those hyperbolas that open up and down versus those that open up left and right.
By looking at the equation, I think we can agree that if x2 equals 0 we've got a problem, since no value of y can satisfy the equation, and in fact x must not be less than 1. Hence, we can already predict that this system opens left and right, as the plot shows (
a2 = b2 = 2
):![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwhXcvk1iWUhKTVNvzwRlYkp4IL6sWv3DAa7FT0LQgrCk-Is1f5hK8VczC7JpjiglJLdRKVs0ScpXfCLj1mqO0eaCYAGbuBP3gpta7_O1QfEyneUtkTY-ab7yRXHyJy4mgqLbMufBeD38s/s320/Screen+shot+2011-06-26+at+11.24.38+AM.png)
The ALEKS review goes on to explain that for this type of equation, the vertices of the hyperbola (points of closest approach to the origin at
h,k
) are equal to and the asymptotes have slopes equal to
This all works great. The problem I had was that the simplest parabola I can think of is:
which doesn't fit the system.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJ2DQXiZ5CC6O6M7vTA-gVWz1u0eVg_gC0TAfOFLb4pEAY47XkHnuE86l7Zs3FFPYmWtL4ok8e4ukExAes_pkRTrtNfyGYcC0ocQTiOAEobEpHKrz1A9AN5G8nJW6DaqccMIjOA9pskipC/s320/Screen+shot+2011-06-26+at+11.26.37+AM.png)
The answer to my confusion is that any parabola may be rotated around its origin in the xy-plane. In 2 selected orientations its equation will have only
x2
and y2
terms (when the vertices are on the x or the y-axis), whereas in 2 other selected orientations it may have only xy
terms (when the vertices are on y= +/- x
. The rest of the time it contains both.This is explained in an excerpt from Stewart's Calculus I found here.
Consider a point
P
in the plane with coordinates x,y
and distance from the origin r
. Now, let's establish a new coordinate system u,v
which is rotated counter-clockwise by the angle θ
. A vector from the origin through P
is rotated an angle φ
with respect to the u axis and φ + θ
with respect to the x-axis. Here's a screenshot from the pdf:![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2YMvT0mdh4mcwORFtqorz3sn4_rZ-ztdEtk-uaHtRceHPDrBxLFfT6PI7mAecfgLXuU2zXTa6BkyGW49q8YSyulzjQnrA7ii3CIVLZMSBL1xlRysqXVQclT_OC26nr-YjP3oKuTuxvShq/s320/Screen+shot+2011-06-26+at+11.21.44+AM.png)
(Note, he uses capital X and Y for the second set of coordinates).
In the
u,v
system, the point P
has coordinates In the
x,y
system the coordinates are:We remember (my post here) that:
Hence:
Now, consider the hyperbola:
Rewrite this in terms of
u,v
and multiply, giving 4 termsIf
θ
equals π/4
, then The two middle terms drop out and leave us with:
If the middle terms don't cancel, we're left with a mixture including some fraction of
u2/2, v2/2, and u times v.
Note: we rotated the coordinate system counter-clockwise, which has the effect of rotating the plot clockwise, when the coordinate system is viewed in standard orientation.