For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. x − 2 2 9 + y 2 16 = 1
For Exercises 9-16, a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola. b. Graph the curve. c. Identify key features of the graph. That is, If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. x − 2 2 9 + y 2 16 = 1
Solution Summary: The author explains the nature of the curve, which represents an ellipse with center (2,0)
a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola.
b. Graph the curve.
c. Identify key features of the graph. That is,
If the equation represents a circle, identify the center and radius.
If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity.
If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity.
If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry.
A factorization A = PDP 1 is not unique. For A=
7 2
-4 1
1
1
5 0
2
1
one factorization is P =
D=
and P-1
30
=
Use this information with D₁
=
to find a matrix P₁ such that
-
-1 -2
0 3
1
-
- 1
05
A-P,D,P
P1
(Type an integer or simplified fraction for each matrix element.)
Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.
30 -1
-
1 0 -1
400
0
0 1
A=
3 4 3
0 1 3
040
3 1 3
0 0
4
1
0
0
003
-1 0 -1
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
A basis for the corresponding eigenspace is {
A. There is one distinct eigenvalue, λ =
B. In ascending order, the two distinct eigenvalues are λ₁
...
=
and 2
=
Bases for the corresponding eigenspaces are {
and ( ), respectively.
C. In ascending order, the three distinct eigenvalues are λ₁ =
=
12/2
=
and 3 = Bases for the corresponding eigenspaces are
{}, }, and {
respectively.
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