Hyperbola Construction

The usual standard form for a hyperbola is
    (x / a)^2 - (y / b)^2 = 1,
with a and b each in the open interval (0, oo).
This equation has the two branches of the hyperbola on the left and right sides of the origin. The asymptotes are the straight lines
    x / a - y / b = 0
    x / a + y / b = 0.
They intersect at the origin. This hyperbola passes through the point
    P(a, 0).
Conversely, it is obvious that if we were given these asymptotes and the point; the foregoing would be the equation of the unique hyperbola.

How can the problem be generalized?

We could translate the point of intersection to (h, k).
Then the equation of the hyperbola becomes
    ((x - h) / a)^2 - ((y - k) / b)^2 = 1.
The asymptotes become
    (x - h) / a - (y - k) / b = 0
    (x - h) / a + (y - k) / b = 0.
The hyperbola passes through the point
    P(a + h, k).

The specified point P(u, v) could be taken as any point that satisfies the equation of the hyperbola. That is, any point for which
((u - h) / a)^2 - ((v - k) / b)^2 = 1.

Finally, we could rotate the coordinates. Pray, see the discussion of the Jacobi rotation at the beginning of the Jacobi algorithm for the eigen pair problem.
Suddenly, the problem has become tedious, to say the least!

Let us state the problem formally.
Problem: Given the pair of intersecting straight lines
a x + b y + c = 0
d x + e y + f = 0
and the point P(u, v), not on either of them.
Find the hyperbola which has these lines as its asymptotes and passes through the given point.
Prove that such a hyperbola exists and that it is unique.

[This is a formalization of a problem assigned in the third semester of Calculus, by an instructor at our college, a few years ago.
Give yourself the two weeks that had been provided by the aforementioned assignment, before reading further to see the answer.]

The traditional method. Employ Linear Algebra. We provide merely an outline.
In principle, while the following computation may be done analytically, usually one would do it only numerically.
Step one. Translation. Pray, see the first chapter.
The two simultaneous equations may be written as the matrix equation
    (a, b; d, e) (x, y)tr = - (c, f)tr.
Let the vector (xi, eta) = (x, y) - (h, k).
Add the vector (h, k), to obtain
    (x, y) = (xi, eta) + (h, k).
Substitute into the foregoing matrix equation, to obtain
    (a, b; d, e) ((xi, eta) + (h, k))tr = - (c, f)tr.
Employ the distributive law of multiplication over addition, to obtain
    (a, b; d, e) (xi, eta)tr + (a, b; d, e) (h, k)tr = - (c, f)tr.
Subtract the second term, to obtain
    (a, b; d, e) (xi, eta)tr = - ((a, b; d, e) (h, k)tr + (c, f)tr).
We want the right-hand side to become zero. To achieve this goal, let the vector (h, k) be given by
    (h, k)tr = - (a, b; d, e)inv (c, f)tr.
The inverse of the matrix exists; because, the determinant of the matrix is not zero as a consequence of the hypothesis that the lines intersect.
Then, the problem has been reduced to that of the two lines
    (a, b; d, e) (xi, eta)tr = 0
and the point P(u - h, v - k).
Step two. Rotation. Pray, see the Jacobi algorithm
The two simultaneous equations are
    a xi + b eta = 0
    d xi + e eta = 0.
Let the two angles be defined as
    phi-1 = Atan(a / b)
    phi-2 = Atan(d / e).
In terms of these angles, the two equations become
    xi sin(phi-1) + eta cos(phi-1) = 0
    xi sin(phi-2) + eta cos(phi-2) = 0.
The addition theorem of the tangents may be employed to express the tangent of the sum of these two angles as a rational algebraic function of the coefficients {a, b, d, e}.
We want to rotate by half the sum. Remember the half-angle formula for the tangent of an angle? That is, by
    alpha = (phi-1 + phi-2) / 2.
Thus, let a new x and y be given by
    (x, y) = (xi, eta) (cos(alpha), sin(alpha); - sin(alpha), cos(alpha)).

After this rotation, we have reduced the problem to that considered at the outset. Hence, we now have the equation for the unique hyperbola.
However, it remains to undo the rotation and the translation, to obtain the equation in the original orientation and location.

I hope that I have not made any stupid mistakes of sign in the foregoing. It is all too easy to do so. :-)

Is there an easier way? Aye, forsooth, there be this ---

[I suggest that you give yourself another two seeks, before proceeding.]

Theorem: Given the pair of intersecting straight lines
    a x + b y + c = 0
    d x + e y + f = 0
and the point P(u, v), not on either of them. Then, the unique hyperbola with these asymptotes and passing through this point is
    (a x + b y + c) (d x + e y + f) = (a u + b v + c) (d u + e v + f).
Proof consists of three observations:
That the curve passes through the point P is obvious.
For the product in the left-side to remain bounded, as x and y increase in magnitude without bound; either one of the factors must approach zero. Hence, we have the two asymptotes.
The equation is a quadratic in x and y; hence, it is a conic section.
The only conic section with asymptotes is a hyperbola. Thus, the equation is that of a hyperbola.