 2.4.1: To complete the square of , you would (add/ subtract) the number .
 2.4.2: Use the Square Root Method to solve the equation 1x  22 2 =
 2.4.3: True or False Every equation of the form has a circle as its graph
 2.4.4: For a circle, the is the distance from the center to any point on t...
 2.4.5: True or False The radius of the circle is 3.
 2.4.6: True or False The center of the circle is . (3, 2) 1x + 322 + 1y ...
 2.4.7: In 710, find the center and radius of each circle. Write the standa...
 2.4.8: In 710, find the center and radius of each circle. Write the standa...
 2.4.9: In 710, find the center and radius of each circle. Write the standa...
 2.4.10: In 710, find the center and radius of each circle. Write the standa...
 2.4.11: In 1120, write the standard form of the equation and the general fo...
 2.4.12: In 1120, write the standard form of the equation and the general fo...
 2.4.13: In 1120, write the standard form of the equation and the general fo...
 2.4.14: In 1120, write the standard form of the equation and the general fo...
 2.4.15: In 1120, write the standard form of the equation and the general fo...
 2.4.16: In 1120, write the standard form of the equation and the general fo...
 2.4.17: In 1120, write the standard form of the equation and the general fo...
 2.4.18: In 1120, write the standard form of the equation and the general fo...
 2.4.19: In 1120, write the standard form of the equation and the general fo...
 2.4.20: In 1120, write the standard form of the equation and the general fo...
 2.4.21: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.22: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.23: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.24: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.25: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.26: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.27: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.28: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.29: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.30: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.31: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.32: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.33: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.34: In 2134, (a) find the center and radius r of each circle; (b) graph...
 2.4.35: In 3542, find the standard form of the equation of each circle Cent...
 2.4.36: In 3542, find the standard form of the equation of each circle Cent...
 2.4.37: In 3542, find the standard form of the equation of each circle Cent...
 2.4.38: In 3542, find the standard form of the equation of each circle Cent...
 2.4.39: In 3542, find the standard form of the equation of each circle With...
 2.4.40: In 3542, find the standard form of the equation of each circle With...
 2.4.41: In 3542, find the standard form of the equation of each circle Cent...
 2.4.42: In 3542, find the standard form of the equation of each circle Cent...
 2.4.43: In 4346, match each graph with the correct equation. (a) (b) (c) (d...
 2.4.44: In 4346, match each graph with the correct equation. (a) (b) (c) (d...
 2.4.45: In 4346, match each graph with the correct equation. (a) (b) (c) (d...
 2.4.46: In 4346, match each graph with the correct equation. (a) (b) (c) (d...
 2.4.47: Find the area of the square in the figure.
 2.4.48: Find the area of the blue shaded region in the figure, assuming the...
 2.4.49: Ferris Wheel The original Ferris wheel was built in 1893 by Pittsbu...
 2.4.50: Ferris Wheel In 2008, the Singapore Flyer opened as the worlds larg...
 2.4.51: Weather Satellites Earth is represented on a map of a portion of th...
 2.4.52: The tangent line to a circle may be defined as the line that inters...
 2.4.53: The Greek Method The Greek method for finding the equation of the t...
 2.4.54: Use the Greek method described in to find an equation of the tangen...
 2.4.55: Refer to 52. The line is tangent to a circle at . The line is tange...
 2.4.56: Find an equation of the line containing the centers of the two circ...
 2.4.57: If a circle of radius 2 is made to roll along the xaxis, what is a...
 2.4.58: If the circumference of a circle is , what is its radius? 6p
 2.4.59: Which of the following equations might have the graph shown? (More ...
 2.4.60: Which of the following equations might have the graph shown? (More ...
 2.4.61: Explain how the center and radius of a circle can be used to graph ...
 2.4.62: What Went Wrong? A student stated that the center and radius of the...
 2.4.63: Center of a Circle Open the Circle: the role of the center applet. ...
 2.4.64: Radius of a Circle Open the Circle: the role of the radius applet. ...
Solutions for Chapter 2.4: Circles
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 2.4: Circles
Get Full SolutionsSince 64 problems in chapter 2.4: Circles have been answered, more than 34495 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: College Algebra, edition: 9. College Algebra was written by and is associated to the ISBN: 9780321716811. Chapter 2.4: Circles includes 64 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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.

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

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!).

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.

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.

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.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

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.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.