 8.1.1: Define an ellipse in terms of its foci.
 8.1.2: Where must the foci of an ellipse lie?
 8.1.3: What special case of the ellipse do we have when the major and mino...
 8.1.4: For the special case mentioned in the previous question, what would...
 8.1.5: What can be said about the symmetry of the graph of an ellipse with...
 8.1.6: For the following exercises, determine whether the given equations ...
 8.1.7: For the following exercises, determine whether the given equations ...
 8.1.8: For the following exercises, determine whether the given equations ...
 8.1.9: For the following exercises, determine whether the given equations ...
 8.1.10: For the following exercises, determine whether the given equations ...
 8.1.11: For the following exercises, write the equation of an ellipse in st...
 8.1.12: For the following exercises, write the equation of an ellipse in st...
 8.1.13: For the following exercises, write the equation of an ellipse in st...
 8.1.14: For the following exercises, write the equation of an ellipse in st...
 8.1.15: For the following exercises, write the equation of an ellipse in st...
 8.1.16: For the following exercises, write the equation of an ellipse in st...
 8.1.17: For the following exercises, write the equation of an ellipse in st...
 8.1.18: For the following exercises, write the equation of an ellipse in st...
 8.1.19: For the following exercises, write the equation of an ellipse in st...
 8.1.20: For the following exercises, write the equation of an ellipse in st...
 8.1.21: For the following exercises, write the equation of an ellipse in st...
 8.1.22: For the following exercises, write the equation of an ellipse in st...
 8.1.23: For the following exercises, write the equation of an ellipse in st...
 8.1.24: For the following exercises, write the equation of an ellipse in st...
 8.1.25: For the following exercises, write the equation of an ellipse in st...
 8.1.26: For the following exercises, write the equation of an ellipse in st...
 8.1.27: For the following exercises, find the foci for the given ellipses. ...
 8.1.28: For the following exercises, find the foci for the given ellipses. ...
 8.1.29: For the following exercises, find the foci for the given ellipses. ...
 8.1.30: For the following exercises, find the foci for the given ellipses. ...
 8.1.31: For the following exercises, find the foci for the given ellipses. ...
 8.1.32: For the following exercises, graph the given ellipses, noting cente...
 8.1.33: For the following exercises, graph the given ellipses, noting cente...
 8.1.34: For the following exercises, graph the given ellipses, noting cente...
 8.1.35: For the following exercises, graph the given ellipses, noting cente...
 8.1.36: For the following exercises, graph the given ellipses, noting cente...
 8.1.37: For the following exercises, graph the given ellipses, noting cente...
 8.1.38: For the following exercises, graph the given ellipses, noting cente...
 8.1.39: For the following exercises, graph the given ellipses, noting cente...
 8.1.40: For the following exercises, graph the given ellipses, noting cente...
 8.1.41: For the following exercises, graph the given ellipses, noting cente...
 8.1.42: For the following exercises, graph the given ellipses, noting cente...
 8.1.43: For the following exercises, graph the given ellipses, noting cente...
 8.1.44: For the following exercises, graph the given ellipses, noting cente...
 8.1.45: For the following exercises, graph the given ellipses, noting cente...
 8.1.46: For the following exercises, use the given information about the gr...
 8.1.47: For the following exercises, use the given information about the gr...
 8.1.48: For the following exercises, use the given information about the gr...
 8.1.49: For the following exercises, use the given information about the gr...
 8.1.50: For the following exercises, use the given information about the gr...
 8.1.51: For the following exercises, use the given information about the gr...
 8.1.52: For the following exercises, given the graph of the ellipse, determ...
 8.1.53: For the following exercises, given the graph of the ellipse, determ...
 8.1.54: For the following exercises, given the graph of the ellipse, determ...
 8.1.55: For the following exercises, given the graph of the ellipse, determ...
 8.1.56: For the following exercises, given the graph of the ellipse, determ...
 8.1.57: For the following exercises, find the area of the ellipse. The area...
 8.1.58: For the following exercises, find the area of the ellipse. The area...
 8.1.59: For the following exercises, find the area of the ellipse. The area...
 8.1.60: For the following exercises, find the area of the ellipse. The area...
 8.1.61: For the following exercises, find the area of the ellipse. The area...
 8.1.62: Find the equation of the ellipse that will just fit inside a box th...
 8.1.63: Find the equation of the ellipse that will just fit inside a box th...
 8.1.64: An arch has the shape of a semiellipse (the top half of an ellipse...
 8.1.65: An arch has the shape of a semiellipse. The arch has a height of 1...
 8.1.66: A bridge is to be built in the shape of a semielliptical arch and i...
 8.1.67: A person in a whispering gallery standing at one focus of the ellip...
 8.1.68: A person is standing 8 feet from the nearest wall in a whispering g...
Solutions for Chapter 8.1: THE ELLIPSE
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 8.1: THE ELLIPSE
Get Full SolutionsSince 68 problems in chapter 8.1: THE ELLIPSE have been answered, more than 34485 students have viewed full stepbystep solutions from this chapter. Chapter 8.1: THE ELLIPSE includes 68 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9781938168383. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 1.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.