 Chapter 10.1: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.2: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.3: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.4: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.5: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.6: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.7: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.8: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.9: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.10: In 110, identify each equation. If it is a parabola, give its verte...
 Chapter 10.11: Parabola; focus at 1 2, 02;directrix the line x = 2
 Chapter 10.12: Hyperbola; center at 10, 02; focus at 10, 42; vertex at 10, 22
 Chapter 10.13: Ellipse; foci at 1 3, 02 and 13, 02; vertex at 14, 02
 Chapter 10.14: Parabola; vertex at 12, 32; focus at 12, 42
 Chapter 10.15: Hyperbola; center at 1 2, 32; focus at 1 4, 32; vertex at 1 3,...
 Chapter 10.16: Ellipse; foci at 1 4, 22 and 1 4, 82; vertex at 1 4, 102
 Chapter 10.17: Center at 1 1, 22; a = 3; c = 4; transverse axis parallel to the x...
 Chapter 10.18: Vertices at 10, 12 and 16, 12; asymptote the line 3y + 2x = 9
 Chapter 10.19: In 19 23, identify each conic without completing the squares and wi...
 Chapter 10.20: In 19 23, identify each conic without completing the squares and wi...
 Chapter 10.21: In 19 23, identify each conic without completing the squares and wi...
 Chapter 10.22: In 19 23, identify each conic without completing the squares and wi...
 Chapter 10.23: In 19 23, identify each conic without completing the squares and wi...
 Chapter 10.24: In 24 26, rotate the axes so that the new equation contains no xyt...
 Chapter 10.25: In 24 26, rotate the axes so that the new equation contains no xyt...
 Chapter 10.26: In 24 26, rotate the axes so that the new equation contains no xyt...
 Chapter 10.27: In 2729, identify the conic that each polar equation represents and...
 Chapter 10.28: In 2729, identify the conic that each polar equation represents and...
 Chapter 10.29: In 2729, identify the conic that each polar equation represents and...
 Chapter 10.30: In 30 and 31, convert each polar equation to a rectangular equation.
 Chapter 10.31: In 30 and 31, convert each polar equation to a rectangular equation.
 Chapter 10.32: In 3234, graph the curve whose parametric equations are given by ha...
 Chapter 10.33: In 3234, graph the curve whose parametric equations are given by ha...
 Chapter 10.34: In 3234, graph the curve whose parametric equations are given by ha...
 Chapter 10.35: Find two different parametric equations for y =  2x + 4.
 Chapter 10.36: Find parametric equations for an object that moves along the ellips...
 Chapter 10.37: Find an equation of the hyperbola whose foci are the vertices of th...
 Chapter 10.38: Describe the collection of points in a plane so that the distance f...
 Chapter 10.39: A searchlight is shaped like a paraboloid of revolution. If a light...
 Chapter 10.40: A bridge is built in the shape of a semielliptical arch. The bridge...
 Chapter 10.41: In a test of their recording devices, a team of seismologists posit...
 Chapter 10.42: Marys train leaves at 7:15 am and accelerates at the rate of 3 mete...
 Chapter 10.43: Drew Brees throws a football with an initial speed of 80 feet per s...
 Chapter 10.44: Formulate a strategy for discussing and graphing an equation of the...
Solutions for Chapter Chapter 10: Analytic Geometry
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter Chapter 10: Analytic Geometry
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6. Since 44 problems in chapter Chapter 10: Analytic Geometry have been answered, more than 59476 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. Chapter Chapter 10: Analytic Geometry includes 44 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Solvable system Ax = b.
The right side b is in the column space of A.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).