Hi Chris
The equation of an hyperbola centered at the origin with the x and y axes as its axes of symmetry can be written as
x^2/a^2 - y^2/b^2 = 1.
Introduce a rotation about the origin and that will produce an xy term.
Shift the centre away from the origin and that will give rise to x and y terms.
(A shift away from the origin, without a rotation, will give rise to x and y terms but no xy term).
The two lines intersect at the point (32/7, 8/7) and this will be the centre of the hyperbola. The transformation X = x - 32/7, Y = y - 8/7, sets up new co-ordinate axes with origin at that point so that the equation of the hyperbola will not contain X and Y terms.
The asymptotes are the lines to which the hyperbola tends as we move further and further away from the origin, that is, as X and Y tend to infinity. Dividing throughout by X^2 allows us to let X and Y tend to infinity, (on the assumption that the ratio Y/X remains finite).
If you multiply the equations of the lines together, you get
60y^2 + 65x^2 - 128xy - 448x +448y + 768 = 0,
and if you graph this, you get, as you should expect, a pair of straight lines, the lines from which the equation was derived.
Now change the value of the constant.
What you should see, is that if if you take the constant to be less that 768, 760 say or smaller, you will get a hyperbola which lies 'outside' the two straight lines, (and by that I mean that it occupies the regions for which the angle between the lines is obtuse).
If on the other hand you take values for the constant greater than 768 you should find that the resulting hyperbola lies between the two lines, ( the regions for which the angle between them is acute).
The fact that the equation of the hyperbola can be derived in this way isn't a coincidence, it should be true in most cases.
(I've just started to think about x = 0, y = 0 which should give rise to xy = 1 as an exception !).
- Bertie
