 103.1: Write an equation for the graph at the right.
 103.2: Write an equation for the orbit of the satellite. Use the center of...
 103.3: Draw a labeled sketch of Earth and the orbit to scale.
 103.4: center (1, 5), radius 2 units
 103.5: endpoints of a diameter at (4, 1) and (4, 5)
 103.6: endpoints of a diameter at (2, 2) and (2, 6)
 103.7: (x  4)2 + (y  1)2 = 9
 103.8: x2 + (y  14)2 = 34
 103.9: (x  4)2 + y2 = _16 25
 103.10: x + _2 3 2 + y  _1 2 2 = 8 9 11.
 103.11: x2 + y2 + 8x  6y = 0
 103.12: x2 + y2 + 4x  8 = 0
 103.13: y O x
 103.14: y O x
 103.15: LANDSCAPING The design of a garden is shown at the right. A pond is...
 103.16: center (0, 3), radius 7 units
 103.17: center (8, 7), radius _1 2 unit
 103.18: endpoints of a diameter at (5, 2) and (3, 6)
 103.19: endpoints of a diameter at (11, 18) and (13, 19)
 103.20: x2 + (y + 2)2 = 4
 103.21: x2 + y2 = 144
 103.22: (x  3)2 + (y  1)2 = 25
 103.23: (x + 3)2 + (y + 7)2 = 81
 103.24: (x  3)2 + y2 = 16
 103.25: (x  3)2 + (y + 7)2 = 50
 103.26: x2 + y2 + 6y = 50  14x
 103.27: x2 + y2  6y  16 = 0
 103.28: x2 + y2 + 2x  10 = 0
 103.29: x2 + y2  18x  18y + 53 = 0
 103.30: x2 + y2 + 9x  8y + 4 = 0
 103.31: x2 + y2  3x + 8y = 20
 103.32: center (8, 9), passes through (21, 22)
 103.33: center  13 , 42 , passes through the origin 3
 103.34: center at (8, 7), tangent to yaxis
 103.35: center at (4, 2), tangent to xaxis
 103.36: center in the first quadrant; tangent to x = 3, x = 5, and the xaxis
 103.37: center in the second quadrant; tangent to y = 1, y = 9, and the y...
 103.38: EARTHQUAKES The Rose Bowl is located about 35 miles west and about ...
 103.39: RADIO The diagram at the right shows the relative locations of some...
 103.40: Solve (x + 3)2 + y2 = 16 for y.
 103.41: What two functions should you enter to graph the given equation?
 103.42: Graph (x + 3)2 + y2 = 16 on a graphing calculator.
 103.43: Solve (x + 3)2 + y2 = 16 for x. What parts of the circle do the two...
 103.44: OPEN ENDED Write an equation for a circle with center at (6, 2).
 103.45: REASONING Write x2 + y2 + 6x  2y  54 = 0 in standard form by comp...
 103.46: FIND THE ERROR Juwan says that the circle with equation (x  4)2 + ...
 103.47: CHALLENGE A circle has its center on the line with equation y = 2x....
 103.48: Writing in Math Use the information about radar equipment on page 5...
 103.49: ACT/SAT What is the center of the circle with equation x2 + y2  10...
 103.50: REVIEW If the surface area of a cube is increased by a factor of 9,...
 103.51: x = 3y2 + 1 5
 103.52: y + 2 = (x  3)2
 103.53: y = x2 + 4x
 103.54: (5, 7), (3, 1)
 103.55: (2, 9), (4, 5)
 103.56: (8, 0), (5, 12)
 103.57: f(x) = x3 + 5x2 + 2x  8 5
 103.58: g(x) = 2x3  9x2 + 7x + 6
 103.59: PHOTOGRAPHY The perimeter of a rectangular picture is 86 inches. Tw...
 103.60: c2 = 132  52
 103.61: c2 = 102  82
 103.62: 7 2 = a2  32 63
 103.63: 42 = 62  b2
Solutions for Chapter 103: Circles
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 103: Circles
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 103: Circles includes 63 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by Patricia and is associated to the ISBN: 9780078738302. Since 63 problems in chapter 103: Circles have been answered, more than 23324 students have viewed full stepbystep solutions from this chapter.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

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.

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!).

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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