About the chord of contact and tangent

This question is not difficult if you know Joachimsthal's notations. Otherwise it is quite tedious
Using Joachimsthal's notations, the equation of the chord is
si = 0
yyi = (1/2)4a(x+xi) where (xi,yi)=(-a,a)
ay=2a(x-a)
y=2(x-a)...(1)
Let the contact point of the chord and the parabola is (x1,y1) (x2,y2)
Now from y²=4ax
x=y²/4a
substitute into (1)
y=2(y²/4a -a)
Simplify
y²-2ay-4a²=0
(y1-y2)^2
=(y1+y2)^2-4y1y2
=4a^2+16a^2
=20a^2
To find (x1-x2)^2
Since
y1^2-y2^2=4a(x1-x2)
(y1+y2)(y1-y2)=4a(x1-x2)
(y1+y2)^2(y1-y2)^2=16a^2(x1-x2)^2
(4a^2)(20a^2)=16a^2(x1-x2)^2
(x1-x2)^2=5a^2
The length of the chord of contact of tangents
=√[(x1-x2)^2+(y1-y2)^2]
=√(5a^2+20a^2)
=√(25a^2)
=5a

2007-10-25 02:42:39 補充：
其實這條是不是pure math題?

2007-10-27 06:29:31 補充：
1 pure math 書無講Joachimsthal's notations﹐所以我有點懷疑這條不是pure math。2 yes