 10.3.1: In Exercises 1 and 2, fill in the blank(s). 1. A _______ is the set...
 10.3.2: In Exercises 1 and 2, fill in the blank(s). 2. The line segment con...
 10.3.3: The form (y k)2 a2 (x h)2 b2 = 1 represents a hyperbola with center...
 10.3.4: How many asymptotoes does a hyperbola have?
 10.3.5: Where do the asymptotes of a hyperbola intersect?
 10.3.6: What type of conic does Ax2 + Cy2 + Dx + Ey + F = 0 represent when ...
 10.3.7: In Exercises 710, match the equation with its graph. [The graphs ar...
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 10.3.10: In Exercises 710, match the equation with its graph. [The graphs ar...
 10.3.11: In Exercises 1120, find the standard form of the equation of the hy...
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 10.3.20: In Exercises 1120, find the standard form of the equation of the hy...
 10.3.21: In Exercises 2130, find the center, vertices, foci, and asymptotes ...
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 10.3.30: In Exercises 2130, find the center, vertices, foci, and asymptotes ...
 10.3.31: In Exercises 31 40, (a) find the standard form of the equation of t...
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 10.3.39: In Exercises 31 40, (a) find the standard form of the equation of t...
 10.3.40: In Exercises 31 40, (a) find the standard form of the equation of t...
 10.3.41: In Exercises 4150, find the standard form of the equation of the hy...
 10.3.42: In Exercises 4150, find the standard form of the equation of the hy...
 10.3.43: In Exercises 4150, find the standard form of the equation of the hy...
 10.3.44: In Exercises 4150, find the standard form of the equation of the hy...
 10.3.45: In Exercises 4150, find the standard form of the equation of the hy...
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 10.3.50: In Exercises 4150, find the standard form of the equation of the hy...
 10.3.51: You and a friend live 4 miles apart (on the same eastwest street) ...
 10.3.52: Three listening stations located at (3300, 0), (3300, 1100), and (3...
 10.3.53: The base for the pendulum of a clock has the shape of a hyperbola. ...
 10.3.54: Long distance radio navigation for aircraft and ships uses synchron...
 10.3.55: A panoramic photo can be taken using a hyperbolic mirror. The camer...
 10.3.56: A hyperbolic mirror (used in some telescopes) has the property that...
 10.3.57: In Exercises 57 66, classify the graph of the equation as a circle,...
 10.3.58: In Exercises 57 66, classify the graph of the equation as a circle,...
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 10.3.67: In Exercises 6770, determine whether the statement is true or false...
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 10.3.71: Consider a hyperbola centered at the origin with a horizontal trans...
 10.3.72: Use the figure to explain why d2 d1 = 2a.
 10.3.73: Find the equation of the hyperbola for any point at which the diffe...
 10.3.74: Show that c2 = a2 + b2 for the equation of the hyperbola x2 a2 y2 b...
 10.3.75: Prove that the graph of the equation Ax2 + Cy2 + Dx + Ey + F = 0 is...
 10.3.76: Given the hyperbolas x2 16 y2 9 = 1 and y2 9 x2 16 = 1 describe any...
 10.3.77: In Exercises 77 82, factor the polynomial completely. 77. x3 16x 78...
 10.3.78: In Exercises 77 82, factor the polynomial completely. 77. x3 16x 78...
 10.3.79: In Exercises 77 82, factor the polynomial completely. 77. x3 16x 78...
 10.3.80: In Exercises 77 82, factor the polynomial completely. 77. x3 16x 78...
 10.3.81: In Exercises 77 82, factor the polynomial completely. 77. x3 16x 78...
 10.3.82: In Exercises 77 82, factor the polynomial completely. 77. x3 16x 78...
Solutions for Chapter 10.3: Topics in Analytic Geometry
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 10.3: Topics in Analytic Geometry
Get Full SolutionsAlgebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. Since 82 problems in chapter 10.3: Topics in Analytic Geometry have been answered, more than 61000 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.3: Topics in Analytic Geometry includes 82 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.