 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.1: Identify the graph of each equation as an ellipse or a hyperbola. D...
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.2: Identify the graph of each equation as an ellipse or a hyperbola. D...
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.3: Identify the graph of each equation as an ellipse or a hyperbola. D...
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.4: Identify the graph of each equation as an ellipse or a hyperbola. D...
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.5: Identify the graph of each equation as an ellipse or a hyperbola. D...
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.6: Identify the graph of each equation as an ellipse or a hyperbola. D...
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.7: Sketch the graph of each equation. See Examples 1 and 2.x24+ y225 = 1
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.8: Sketch the graph of each equation. See Examples 1 and 2.x216+ y29 = 1
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.9: Sketch the graph of each equation. See Examples 1 and 2.x29+ y2 = 1
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.10: Sketch the graph of each equation. See Examples 1 and 2.x2 + y24 = 1
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.11: Sketch the graph of each equation. See Examples 1 and 2.9x2 + y2 = 36
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.12: Sketch the graph of each equation. See Examples 1 and 2.x2 + 4y2 = 16
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.13: Sketch the graph of each equation. See Examples 1 and 2.4x2 + 25y2 ...
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.14: Sketch the graph of each equation. See Examples 1 and 2.36x2 + y2 = 36
 10.2.Integrated Review: Identify whether each equation, when graphed, will be a parabola, c...
 10.2.15: Sketch the graph of each equation. See Example 31x + 12236+ 1y  22...
 10.2.16: Sketch the graph of each equation. See Example 31x  3229+ 1y + 322...
 10.2.17: Sketch the graph of each equation. See Example 31x  1224+ 1y  122...
 10.2.18: Sketch the graph of each equation. See Example 31x + 32216+ 1y + 22...
 10.2.19: Sketch the graph of each equation. See Examples 4 and 5.x24  y29 = 1
 10.2.20: Sketch the graph of each equation. See Examples 4 and 5.x236  y236...
 10.2.21: Sketch the graph of each equation. See Examples 4 and 5.y225  x216...
 10.2.22: Sketch the graph of each equation. See Examples 4 and 5.y225  x249...
 10.2.23: Sketch the graph of each equation. See Examples 4 and 5.x2  4y2 = 16
 10.2.24: Sketch the graph of each equation. See Examples 4 and 5.4x2  y2 = 36
 10.2.25: Sketch the graph of each equation. See Examples 4 and 5.16y2  x2 = 16
 10.2.26: Sketch the graph of each equation. See Examples 4 and 5.4y2  25x2 ...
 10.2.27: Graph each equation. See Examples 1 through 5.y236 = 1  x2
 10.2.28: Graph each equation. See Examples 1 through 5.x236 = 1  y2
 10.2.29: Graph each equation. See Examples 1 through 5.41x  122 + 91y + 222...
 10.2.30: Graph each equation. See Examples 1 through 5.251x + 322 + 41y  32...
 10.2.31: Graph each equation. See Examples 1 through 5.8x2 + 2y2 = 32
 10.2.32: Graph each equation. See Examples 1 through 5.3x2 + 12y2 = 48
 10.2.33: Graph each equation. See Examples 1 through 5.25x2  y2 = 25
 10.2.34: Graph each equation. See Examples 1 through 5.x2  9y2 = 9
 10.2.35: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.36: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.37: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.38: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.39: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.40: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.41: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.42: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.43: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.44: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.45: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.46: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.47: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.48: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.49: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.50: Identify whether each equation, when graphed, will be a parabola,ci...
 10.2.51: Perform the indicated operations. See Sections 5.1 and 5.3.12x321 ...
 10.2.52: Perform the indicated operations. See Sections 5.1 and 5.3.2x3  4x3
 10.2.53: Perform the indicated operations. See Sections 5.1 and 5.3.5x2 + x2
 10.2.54: Perform the indicated operations. See Sections 5.1 and 5.3.1 5x221x22
 10.2.55: The graph of each equation is an ellipse. Determine which distance ...
 10.2.56: The graph of each equation is an ellipse. Determine which distance ...
 10.2.57: The graph of each equation is an ellipse. Determine which distance ...
 10.2.58: The graph of each equation is an ellipse. Determine which distance ...
 10.2.59: If you are given a list of equations of circles, parabolas, ellipse...
 10.2.60: We know that x2 + y2 = 25 is the equation of a circle. Rewrite the ...
 10.2.61: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.62: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.63: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.64: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.65: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.66: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.67: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.68: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.69: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.70: The orbits of stars, planets, comets, asteroids, and satellites all...
 10.2.71: A planets orbit about the sun can be described as an ellipse. Consi...
 10.2.72: Comets orbit the sun in elongated ellipses. Consider the sunas the ...
 10.2.73: Find the center of the path of the comet.Use a graphing calculator ...
 10.2.74: Find the center of the path of the comet.Use a graphing calculator ...
 10.2.75: For Exercises 75 through 80, see the example below.ExampleSketch th...
 10.2.76: For Exercises 75 through 80, see the example below.ExampleSketch th...
 10.2.77: For Exercises 75 through 80, see the example below.ExampleSketch th...
 10.2.78: For Exercises 75 through 80, see the example below.ExampleSketch th...
 10.2.79: For Exercises 75 through 80, see the example below.ExampleSketch th...
 10.2.80: For Exercises 75 through 80, see the example below.ExampleSketch th...
Solutions for Chapter 10.2: The Ellipse and the Hyperbola
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 10.2: The Ellipse and the Hyperbola
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046. Chapter 10.2: The Ellipse and the Hyperbola includes 95 full stepbystep solutions. Since 95 problems in chapter 10.2: The Ellipse and the Hyperbola have been answered, more than 62311 students have viewed full stepbystep solutions from this chapter.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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).

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.

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 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!).

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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 .

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

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).