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):
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.
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
y2terms (when the vertices are on the x or the y-axis), whereas in 2 other selected orientations it may have only
xyterms (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
Pin the plane with coordinates
x,yand distance from the origin
r. Now, let's establish a new coordinate system
u,vwhich is rotated counter-clockwise by the angle
θ. A vector from the origin through
Pis rotated an angle
φwith respect to the u axis and
φ + θwith respect to the x-axis. Here's a screenshot from the pdf:
(Note, he uses capital X and Y for the second set of coordinates).
u,vsystem, the point
x,ysystem the coordinates are:
We remember (my post here) that:
Now, consider the hyperbola:
Rewrite this in terms of
u,vand multiply, giving 4 terms
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.