Page 407 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Clearly for P(x , y ) = P(x, y), (7) reduces to (6). An important observation is that s is symmetrical
j
j
i
ij
i
in its indices:
s = s . ...(8)
ji
ij
The last of Joachimsthal’s conventions brings the first whiff of an indication as to how useful the
notations may be. In s both P(x , y ) and P(x, y) are quite generic. The indices are only needed to
ij
j
j
i
i
distinguish between two points. But if we omit the indices from one of them, the points will be
as distinct as before. One additional convention accommodates this case: for points P(x , y ) and
i
i
P(x, y) we write (7) with one index only,
s = Ax x + B(x y + xy ) + Cy y + F(x + x) + G(y + y) + H. ...(9)
i
i
i
i
i
i
i
The curious thing about (9) is that, although s was probably perceived as a number, s appears
ij
i
to dependent on “variable” x and y and thus is mostly perceived as a function of these variables.
As a function of x and y, (9) is linear, i.e. of first degree, so that s = 0 is an equation of a straight
i
line. What straight line is it? How does it relate to the conic (1)? The beauty of Joachimsthal’s
notations is that the relation between s = 0 and s = 0 is quite transparent.
i
Theorem 11
Let point P(x , y ) lie on the conic s = 0. In other words, assume that s = 0. Then s = 0 is an equation
ii
i
i
i
of the line tangent to s = 0 at P(x , y ).
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i
Proof
Any point P(x, y) on the line through two distinct points P(x , y ) and P(x , y ) is a linear
2
1
1
2
combination of the two points:
.
.
P(x, y) = t P(x , x ) + (1 - t) P(x , x ), ...(10)
2
1
2
1
which is just a parametric equation of the straight line. Substitute (10) into (2). The exercise may
be a little tedious but is quite straightforward. The result is a quadratic expression in t:
.
2 .
s(t) = t (s + s - 2s ) + 2t (s - s ) + s . ...(11)
12
22
12
22
22
11
Line (10) and conic (1) will have 0, 1, or 2 common points depending on the number of roots of
the quadratic equation s(t) = 0, which is determined by the value of the discriminant
.
.
D= (s - s ) - (s + s - 2s ) s = s - s s . ...(12)
2
2
22
22
22
12
12
22
11
12
11
The line is tangent to the conic if the quadratic equation has two equal roots, i.e. when D = 0, or
.
s = s s . ...(13)
2
12
22
11
This is an interesting identity valid for any line tangent to the conic, with P(x , y ) and P(x , y )
2
1
1
2
chosen arbitrarily on the line. We can use this arbitrariness to our advantage. Indeed, what could
be more natural in these circumstances than picking up the point of tangency. Let’s P(x , y ) be
1
1
such a point. This in particular means that the point lies on the conic so that, according to (5),
s = 0. But then (12) implies s = 0. Let’s say this again: If P(x , y ) is the point of tangency of a
11
1
12
1
conic and a line through another arbitrary point P(x , y ) then
2
2
s = 0. ...(14)
12
Now, since this is true for any point P(x , y ) on the tangent at P(x , y ) we may as well drop the
2
2
1
1
index. The conclusion just drops into our lap: the tangent to a conic s = 0 at point P(x , y ) on the
1
1
conic is given by
s = 0, ...(15)
1
which proves the theorem.
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