 10.1.1: The distance, d, between the points (x1, y1) and (x2, y2) in the re...
 10.1.2: The midpoint of a line segment whose endpoints are (x1, y1) and (x2...
 10.1.3: The set of all points in a plane that are equidistant from a fixed ...
 10.1.4: The standard form of the equation of a circle with center (h, k) an...
 10.1.5: The equation x2 + y2 + Dx + Ey + F = 0 is called the ______________...
 10.1.6: In the equation (x2 + 4x ) + (y2  8y ), we complete the square on ...
 10.1.7: In Exercises 118, find the distance between each pair of points. If...
 10.1.8: In Exercises 118, find the distance between each pair of points. If...
 10.1.9: In Exercises 118, find the distance between each pair of points. If...
 10.1.10: In Exercises 118, find the distance between each pair of points. If...
 10.1.11: In Exercises 118, find the distance between each pair of points. If...
 10.1.12: In Exercises 118, find the distance between each pair of points. If...
 10.1.13: In Exercises 118, find the distance between each pair of points. If...
 10.1.14: In Exercises 118, find the distance between each pair of points. If...
 10.1.15: In Exercises 118, find the distance between each pair of points. If...
 10.1.16: In Exercises 118, find the distance between each pair of points. If...
 10.1.17: In Exercises 118, find the distance between each pair of points. If...
 10.1.18: In Exercises 118, find the distance between each pair of points. If...
 10.1.19: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.20: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.21: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.22: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.23: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.24: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.25: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.26: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.27: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.28: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.29: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.30: In Exercises 1930, find the midpoint of the line segment with the g...
 10.1.31: In Exercises 3140, write the standard form of the equation of the c...
 10.1.32: In Exercises 3140, write the standard form of the equation of the c...
 10.1.33: In Exercises 3140, write the standard form of the equation of the c...
 10.1.34: In Exercises 3140, write the standard form of the equation of the c...
 10.1.35: In Exercises 3140, write the standard form of the equation of the c...
 10.1.36: In Exercises 3140, write the standard form of the equation of the c...
 10.1.37: In Exercises 3140, write the standard form of the equation of the c...
 10.1.38: In Exercises 3140, write the standard form of the equation of the c...
 10.1.39: In Exercises 3140, write the standard form of the equation of the c...
 10.1.40: In Exercises 3140, write the standard form of the equation of the c...
 10.1.41: In Exercises 4148, give the center and radius of the circle describ...
 10.1.42: In Exercises 4148, give the center and radius of the circle describ...
 10.1.43: In Exercises 4148, give the center and radius of the circle describ...
 10.1.44: In Exercises 4148, give the center and radius of the circle describ...
 10.1.45: In Exercises 4148, give the center and radius of the circle describ...
 10.1.46: In Exercises 4148, give the center and radius of the circle describ...
 10.1.47: In Exercises 4148, give the center and radius of the circle describ...
 10.1.48: In Exercises 4148, give the center and radius of the circle describ...
 10.1.49: In Exercises 4956, complete the square and write the equation in st...
 10.1.50: In Exercises 4956, complete the square and write the equation in st...
 10.1.51: In Exercises 4956, complete the square and write the equation in st...
 10.1.52: In Exercises 4956, complete the square and write the equation in st...
 10.1.53: In Exercises 4956, complete the square and write the equation in st...
 10.1.54: In Exercises 4956, complete the square and write the equation in st...
 10.1.55: In Exercises 4956, complete the square and write the equation in st...
 10.1.56: In Exercises 4956, complete the square and write the equation in st...
 10.1.57: In Exercises 5760, find the solution set for each system by graphin...
 10.1.58: In Exercises 5760, find the solution set for each system by graphin...
 10.1.59: In Exercises 5760, find the solution set for each system by graphin...
 10.1.60: In Exercises 5760, find the solution set for each system by graphin...
 10.1.61: In Exercises 6164, write the standard form of the equation of the c...
 10.1.62: In Exercises 6164, write the standard form of the equation of the c...
 10.1.63: In Exercises 6164, write the standard form of the equation of the c...
 10.1.64: In Exercises 6164, write the standard form of the equation of the c...
 10.1.65: In Exercises 6566, a line segment through the center of each circle...
 10.1.66: In Exercises 6566, a line segment through the center of each circle...
 10.1.67: In Exercises 6768, use the information at the bottom of the previou...
 10.1.68: In Exercises 6768, use the information at the bottom of the previou...
 10.1.69: A rectangular coordinate system with coordinates in miles is placed...
 10.1.70: The Ferris wheel in the figure has a radius of 68 feet. The clearan...
 10.1.71: In your own words, describe how to find the distance between two po...
 10.1.72: In your own words, describe how to find the midpoint of a line segm...
 10.1.73: What is a circle? Without using variables, describe how the definit...
 10.1.74: . Give an example of a circles equation in standard form. Describe ...
 10.1.75: How is the standard form of a circles equation obtained from its ge...
 10.1.76: Does (x  3) 2 + (y  5) 2 = 0 represent the equation of a circle? ...
 10.1.77: Does (x  3) 2 + (y  5) 2 = 25 represent the equation of a circle...
 10.1.78: In Exercises 7880, use a graphing utility to graph each circle whos...
 10.1.79: In Exercises 7880, use a graphing utility to graph each circle whos...
 10.1.80: In Exercises 7880, use a graphing utility to graph each circle whos...
 10.1.81: Make Sense? In Exercises 8184, determine whether each statement mak...
 10.1.82: Make Sense? In Exercises 8184, determine whether each statement mak...
 10.1.83: Make Sense? In Exercises 8184, determine whether each statement mak...
 10.1.84: Make Sense? In Exercises 8184, determine whether each statement mak...
 10.1.85: In Exercises 8588, determine whether each statement is true or fals...
 10.1.86: In Exercises 8588, determine whether each statement is true or fals...
 10.1.87: In Exercises 8588, determine whether each statement is true or fals...
 10.1.88: In Exercises 8588, determine whether each statement is true or fals...
 10.1.89: Show that the points A(1, 1 + d), B(3, 3 + d), and C(6, 6 + d) are ...
 10.1.90: Prove the midpoint formula by using the following procedure. a. Sho...
 10.1.91: Find all points with y@coordinate 2 so that the distance between (x...
 10.1.92: Find the area of the doughnutshaped region bounded by the graphs o...
 10.1.93: A tangent line to a circle is a line that intersects the circle at ...
 10.1.94: If f(x) = x2  2 and g(x) = 3x + 4, find f(g(x)) and g(f(x)). (Sect...
 10.1.95: Solve: 2x = 27x  3 + 3. (Section 7.6, Example 3)
 10.1.96: Solve: 2x  5 6 10. (Section 4.3, Example 4)
 10.1.97: Exercises 9799 will help you prepare for the material covered in th...
 10.1.98: Exercises 9799 will help you prepare for the material covered in th...
 10.1.99: Exercises 9799 will help you prepare for the material covered in th...
Solutions for Chapter 10.1: Distance and Midpoint Formulas; Circles
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 10.1: Distance and Midpoint Formulas; Circles
Get Full SolutionsIntermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934. Since 99 problems in chapter 10.1: Distance and Midpoint Formulas; Circles have been answered, more than 13361 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 10.1: Distance and Midpoint Formulas; Circles includes 99 full stepbystep solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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