 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 36759 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.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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.

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

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)ยท(b  Ax) = o.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

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).

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.