Butterflies in Hyperbola

Sidney H. Kung

We give an analytic proof of the Butterfly Theorem for hyperbolas.

Theorem

Let M(0, k) be the midpoint of a chord AB parallel to the major axis of a hyperbola. Through M two other chords CD and EF are drawn. ED cuts AB at P and CF cuts AB at Q. Then M is also the midpoint of PQ.

Proof

Through M, we introduce a new set of x-y-axes. Then the equation of the hyperbola is

(1)


-
(y + k)²

=1

or,

(1') - (y + k)² - b² = 0.

Assume that the coordinates of the related points are as follows: M(0, 0), C(x1, y1), D(x2, y2), E(x3, y3), F(x4, y4), P(p, 0), Q(q, 0); and that the equations of lines DC and EF are y = m1x and y = m2x, respectively.

Substituting m1x into (1') gives

(2) (- m1²) - 2m1kx - (b² + k²) = 0.

The roots, x1 and x2, of equation (2) are the x-coordinates of C and D, where

x1 + x2=
-
- 2m1ka²

- m1²a²
    and    
x1
x2=
- a²(b² + k²)

- m1²a²

Dividing these, we get

m1x1x2

x1 + x2
=
- (b² + k²)

2k

Similarly, by substituting y = m2x into (2) we will get

m2x3x4

x3 + x4
=
- (b² + k²)

2k

also (x3 and x4 are x-coordinates of E and F). Thus,

(*)
m1x1x2

x1 + x2
=
m2x3x4

x3 + x4

which can further be rearranged (see note (I) below) to become

(3)
- x2x3

m1x2 - m2x3
=
x1x4

m1x1 - m2x4

Observe that C, Q, F are collinear. The slope of QC and that of FQ are equal. So,

y1

x1 - q
=
- y4

q - x4

Hence,

(4)
q - x1

q - x4
=
y1

y4
=
m1x1

m2x4

q ≠ x1 and q ≠ x4.

Solving (4) for q, we have

q=
(m1 - m2) x1x4

m1x1 - m2x4

In a similar manner, by equating the slopes of PE and DP we can get

p=
(m1 - m2) x2x3

m1x2 - m2x3

Now by comparing p and q, and taking into account (3), it is easy to see that |p| = |q|. Therefore, MP = MQ.

Note

  1. Derivation of (3): cross-multiplying and rearranging,

     m1x1x2(x3 + x4)= m2x3x4(x1 + x2)
     m1x1x2x3 - m2x3x2x4= m2x3x4x1 - m1x1x2x4
     x2x3(m1x1 - m2x4)= x1x4(m2x3 - m1x2),

    implying (3).

  2. The above results are valid for parabolas. To verify, one may use the equation x² = p(y + k), p, k > 0, to start.

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