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10 0
01 1
01 1
Ê
ˆ
Á
Á
˜
˜
M
=
GFE
=
-
,
Ë
¯
then MAM T is the diagonal matrix A 3 , that is, our conic is projectively equivalent to
the unit circle
2
2
xy
+=,
1
and we are done.
Example 3.6.1.2 leads us to some observations about the relationship between a
conic in R 2 and the associated projective conic in P 2 . Consider the solutions to equa-
tion (3.64). One of the solutions (in P 2 ) is the ideal point which has z = 0. Substitut-
ing this value into (3.64) defines the line x = 0 in R 2 . In other words, the parabola y
= x 2 corresponds to the conic in P 2 , which contains the same real points and has one
additional ideal point corresponding to the line x = 0. As another example, consider
the hyperbola
2
2
x
a
y
b
-=.
1
(3.65)
2
2
The homogeneous equation for this conic is
2
2
x
a
y
b
2
--=.
z
0
(3.66)
2
2
The ideal points with z = 0 lead to the equations
b
a x
y
=
(3.67a)
and
b
a x
y
=-
(3.67b)
which define two lines in R 2 . It follows that the conic in P 2 defined by (3.66) is topo-
logically a circle that consists of the points defined by the real roots of equation (3.65)
together with two extra (ideal) points associated to the lines in (3.67a) and (3.67b).
Intuitively, if we were to walk along points (x,y) on the curve (3.65) where these points
approach either (+•,+•) or (-•,-•) we would in either case approach the ideal point
associated to the line defined by equation (3.67a). Letting x and y approach either
(-•,+•) or (+•,-•) would bring us to the ideal point associated to the line defined by
equation (3.67b).
 
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