 7.3.1: The distance d from to is _____.
 7.3.2: To complete the square of add .
 7.3.3: Find the intercepts of the equation y2 = 16  4x2.
 7.3.4: The point that is symmetric with respect to the yaxis to the point...
 7.3.5: To graph shift the graph of to the (left/right) _____ unit(s) and t...
 7.3.6: The standard equation of a circle with center at and radius 1 is __...
 7.3.7: A(n) is the collection of all points in the plane the sum of whose ...
 7.3.8: For an ellipse, the foci lie on a line called the axis.
 7.3.9: For the ellipse the vertices are the points and .
 7.3.10: For the ellipse the value of a is , the value of b is , and the maj...
 7.3.11: If the center of an ellipse is the major axis is parallel to the x...
 7.3.12: If the foci of an ellipse are and , then the coordinates of the cen...
 7.3.13: In 1316, the graph of an ellipse is given. Match each graph to its ...
 7.3.14: In 1316, the graph of an ellipse is given. Match each graph to its ...
 7.3.15: In 1316, the graph of an ellipse is given. Match each graph to its ...
 7.3.16: In 1316, the graph of an ellipse is given. Match each graph to its ...
 7.3.17: x2 25 + y2 4 = 1
 7.3.18: x2 9 + y2 4 = 1
 7.3.19: x2 9 + y2 25 = 1
 7.3.20: x2 + y2 16 = 1
 7.3.21: 4x2 + y2 = 16
 7.3.22: x2 + 9y2 = 18
 7.3.23: 4y2 + x2 = 8
 7.3.24: 4y2 + 9x2 = 36
 7.3.25: x2 + y2 = 16
 7.3.26: x2 + y2 = 4
 7.3.27: Center at focus at vertex at 10, 02; 13, 02; 15, 02
 7.3.28: Center at focus at vertex at 10, 02; 11, 02; 13, 02
 7.3.29: Center at focus at vertex at 10, 02; 10, 42; 10, 52
 7.3.30: Center at focus at vertex at 10, 02; 10, 12; 10, 22
 7.3.31: Foci at length of the major axis is 6
 7.3.32: Foci at length of the major axis is 8
 7.3.33: Focus at vertices at 14, 02; 1;5, 02
 7.3.34: Focus at vertices at 10, 42; 10, ;82
 7.3.35: Foci at 10, ;32; xintercepts are ;2
 7.3.36: Vertices at 1;4, 02; yintercepts are ;1
 7.3.37: Center at vertex at 10, 02; 10, 42; b = 1
 7.3.38: Vertices at 1;5, 02; c = 2
 7.3.39: In 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1,...
 7.3.40: In 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1,...
 7.3.41: In 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1,...
 7.3.42: In 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1,...
 7.3.43: 1x  322 4 + 1y + 122 9 = 1
 7.3.44: 1x + 422 9 + 1y + 222 4 = 1
 7.3.45: 1x + 522 + 41y  422 = 16
 7.3.46: 91x  322 + 1y + 222 = 18
 7.3.47: x2 + 4x + 4y2  8y + 4 = 0
 7.3.48: x2 + 3y2  12y + 9 = 0
 7.3.49: 2x2 + 3y2  8x + 6y + 5 = 0 5
 7.3.50: 4x2 + 3y2 + 8x  6y = 5
 7.3.51: 9x2 + 4y2  18x + 16y  11 = 0
 7.3.52: x2 + 9y2 + 6x  18y + 9 = 0
 7.3.53: 4x2 + y2 + 4y = 0
 7.3.54: 9x2 + y2  18x = 0
 7.3.55: Center at vertex at focus at 12, 22; 17, 22; 14, 22
 7.3.56: Center at vertex at focus at 13, 12; 13, 32; 13, 02
 7.3.57: Vertices at and focus at 14, 32 14, 92; 14, 82
 7.3.58: Foci at and vertex at 11, 22 13, 22; 14, 22
 7.3.59: Foci at and length of the major axis is 8
 7.3.60: Vertices at and 12, 52 12, 12; c = 2
 7.3.61: Center at focus at contains the point 11, 22; 14, 22; 11, 32
 7.3.62: Center at focus at contains the point 11, 22; 11, 42; 12, 22
 7.3.63: Center at vertex at contains the point 11, 22; 14, 22; 11, 52
 7.3.64: Center at vertex at contains the point 11 + 23, 32
 7.3.65: f1x2 = 416  4x2
 7.3.66: f1x2 = 49  9x2
 7.3.67: f1x2 = 464  16x2 6
 7.3.68: f1x2 = 464  16x2 6
 7.3.69: Semielliptical Arch Bridge An arch in the shape of the upper half o...
 7.3.70: Semielliptical Arch Bridge The arch of a bridge is a semiellipse wi...
 7.3.71: Whispering Gallery A hall 100 feet in length is to be designed as a...
 7.3.72: Whispering Gallery Jim, standing at one focus of a whispering galle...
 7.3.73: Semielliptical Arch Bridge A bridge is built in the shape of a semi...
 7.3.74: Semielliptical Arch Bridge A bridge is to be built in the shape of ...
 7.3.75: Racetrack Design Consult the figure. A racetrack is in the shape of...
 7.3.76: Semielliptical Arch Bridge An arch for a bridge over a highway is i...
 7.3.77: Installing a Vent Pipe A homeowner is putting in a fireplace that h...
 7.3.78: Volume of a Football A football is in the shape of a prolate sphero...
 7.3.79: Earth The mean distance of Earth from the Sun is 93 million miles. ...
 7.3.80: Mars The mean distance of Mars from the Sun is 142 million miles. I...
 7.3.81: Jupiter The aphelion of Jupiter is 507 million miles. If the distan...
 7.3.82: Pluto The perihelion of Pluto is 4551 million miles, and the distan...
 7.3.83: Show that an equation of the form Ax2 + Cy2 + F = 0, A Z 0, C Z 0, ...
 7.3.84: Show that the graph of an equation of the form where A and C are of...
 7.3.85: The eccentricity e of an ellipse is defined as the number where a i...
Solutions for Chapter 7.3: The Ellipse
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 7.3: The Ellipse
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra, edition: 9. College Algebra was written by and is associated to the ISBN: 9780321716811. Since 85 problems in chapter 7.3: The Ellipse have been answered, more than 32694 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.3: The Ellipse includes 85 full stepbystep solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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