Math 4800/6080. Week Seven Starter
1. Here is another point of view on plane conics. Show that:

 
x
a b c



x y z
b d e
y =0
c e f
z
is the equation of a plane conic. What is this equation?
2. Find matrices for each conic type: ellipse, hyperbola, parabola,
intersecting lines, parallel lines, one line, one point, empty set.
3. Which of the matrices from Problem 2 has determinant zero?


a b c
det  b d e  = 0
c e f
4. Every symmetric matrix can be converted to



1 0
a b c
T 


0 2
b d e A=
A
0 0
c e f
a diagonal matrix:

0
0 
3
with a suitable invertible 3 × 3 matrix A where the triple (1 , 2 , 3 ) is
one of the following:
(i) (1, 1, 1)
(ii) (−1, 1, 1)
(iii) (−1, −1, 1)
(iv) (−1, −1, −1)
(iv) (1, 1, 0)
(v) (1, −1, 0)
(vi) (−1, −1, 0)
(vii) (1, 0, 0)
(viii) (−1, 0, 0)
(ix) (0, 0, 0)
For each of these diagonal matrices, find the corresponding conic.
Conclude that a conic is a pair of lines or a single line if and only if the
determinant of the matrix is zero. It also follows that all hyperbolas,
ellipses and parabolas are taken to the unit circle under an appropriate
projective linear transformation.
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