Page 347 - 35Linear Algebra

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(e) Is there a direction in which displacing the bridge yields no force? If
so give a vector in that direction. Briefly evaluate the quality of this
bridge design.
12. Conic Sections: The equation for the most general conic section is given by

2
2
ax + 2bxy + dy + 2cx + 2ey + f = 0 .
Our aim is to analyze the solutions to this equation using matrices.

(a) Rewrite the above quadratic equation as one of the form

T
T
T
X MX + X C + C X + f = 0

x
T
relating an unknown column vector X = , its transpose X , a
y
2 × 2 matrix M, a constant column vector C and the constant f.
(b) Does your matrix M obey any special properties? Find its eigenvalues.
You may call your answers λ and µ for the rest of the problem to save
writing.
For the rest of this problem we will focus on central conics for
which the matrix M is invertible.
(c) Your equation in part (a) above should be be quadratic in X. Recall
2
that if m 6= 0, the quadratic equation mx + 2cx + f = 0 can be
rewritten by completing the square

c c
2 2
m x + = − f .
m m
Being very careful that you are now dealing with matrices, use the
same trick to rewrite your answer to part (a) in the form
T
Y MY = g.

Make sure you give formulas for the new unknown column vector Y
and constant g in terms of X, M, C and f. You need not multiply out
any of the matrix expressions you find.
If all has gone well, you have found a way to shift coordinates
for the original conic equation to a new coordinate system
with its origin at the center of symmetry. Our next aim is
to rotate the coordinate axes to produce a readily recognizable
equation.

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