Sunday, June 26, 2011

Different views, same hyperbola

In the algebra study guide that my son is using (ALEKS) they set up a hyperbola in two "standard" forms as:

(x-h)2/a2 - (y-k)2/b2 = 1
(y-k)2/b2 - (x-h)2/a2 = 1

or more generally:

(x-h)2/a2 - (y-k)2/b2 = +/- 1

Let's simplify our lives and consider an example centered at the origin:

x2/a2 - y2/b2 = 1

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

h +/- a, k

and the asymptotes have slopes equal to

+/- b/a

This all works great. The problem I had was that the simplest parabola I can think of is:

xy = 1

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 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:

(Note, he uses capital X and Y for the second set of coordinates).

In the u,v system, the point P has coordinates

u = r cos(φ)
v = r sin(φ)

In the x,y system the coordinates are:

x = r cos(φ + θ)
y = r sin(φ + θ)

We remember (my post here) that:

sin(s+t) = sin s cos t + cos s sin t
cos(s+t) = cos s cos t - sin s sin t


x = r cos(φ) cos(θ) - r sin(φ) sin(θ)
= u cos(θ) - v sin(θ)

y = r sin(φ) cos(θ) + r cos(φ) sin(θ)
= v cos(&theta) + u sin(θ)
= u sin(&theta) + v cos(θ)

Now, consider the hyperbola:

xy = 1

Rewrite this in terms of u,v and multiply, giving 4 terms

+ u cos(θ) u sin(θ)
+ u cos(θ) v cos(θ)
- u sin(θ) v sin(θ)
- v sin(θ) v cos(θ)

If θ equals π/4, then

cos(θ) = sin(θ) = 1/√2

The two middle terms drop out and leave us with:

u2/2 - v2/2 = 1

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.