 4.3.1: A ________ is the intersection of a plane and a doublenapped cone.
 4.3.2: The equation is the standard form of the equation of a ________ wit...
 4.3.3: A ________ is the set of all points in a plane that are equidistant...
 4.3.4: The ________ of a parabola is the midpoint between the focus and th...
 4.3.5: The line that passes through the focus and the vertex of a parabola...
 4.3.6: An ________ is the set of all points in a plane, the sum of whose d...
 4.3.7: The chord joining the vertices of an ellipse is called the ________...
 4.3.8: The chord perpendicular to the major axis at the center of an ellip...
 4.3.9: A ________ is the set of all points in a plane, the difference of w...
 4.3.10: The line segment connecting the vertices of a hyperbola is called t...
 4.3.11: x 2 2y 2 2y
 4.3.12: x x 2 2y 2
 4.3.13: y 2 2x 2 2x
 4.3.14: y y 2 2x 2
 4.3.15: 9x 2 9 2 y 2 9
 4.3.16: x 2 9y 9x 2 9
 4.3.17: 9x 2 9 2 y 2 9
 4.3.18: y2 9x 9x 2 9 2
 4.3.19: x 2 16 2 y2 25
 4.3.20: y x 2 16
 4.3.21: y 1 2x 2
 4.3.22: y 4x 2 y
 4.3.23: y 2 3x 2 6x
 4.3.24: y y 2 3x
 4.3.25: x 2 0 2 12y 0
 4.3.26: y x 2 0
 4.3.27: Focus:
 4.3.28: Focus:
 4.3.29: Focus:
 4.3.30: Focus:
 4.3.31: Directrix
 4.3.32: Directrix:
 4.3.33: Directrix:
 4.3.34: Directrix:
 4.3.35: Passes through the point horizontal axis
 4.3.36: Passes through the point vertical axis
 4.3.37: Passes through the point vertical axis
 4.3.38: Passes through the point horizontal axis
 4.3.39: 4 2 2 4 2 4 6 (3, 6) y
 4.3.40: 8 4 4 8 8 (2, 6) y
 4.3.41: 4 6 6 (5, 3) 2 4 2 4 6 8 10 y
 4.3.42: ( 8, 4) 12 4 4 8 8 y
 4.3.43: FLASHLIGHT The light bulb in a flashlight is at the focus of the pa...
 4.3.44: SATELLITE ANTENNA Write an equation for a cross section of the para...
 4.3.45: SUSPENSION BRIDGE Each cable of the Golden Gate Bridge is suspended...
 4.3.46: BEAM DEFLECTION A simply supported beam (see figure) is 64 feet lon...
 4.3.47: x2 25 y 2 16 1
 4.3.48: x2 121 y 2 144 1
 4.3.49: x2 25 9 y2 16 9 1
 4.3.50: x 2 4 y 2 1 4 1
 4.3.51: x 2 36 y 2 7 1
 4.3.52: x 2 28 y 2 64 1
 4.3.53: 4x 2 36 2 y2 1
 4.3.54: 4x2 9y 4x 2 36
 4.3.55: 1 16x2 1 81y2 1
 4.3.56: 1 100x2 1 49y 2 1
 4.3.57: 2 2 4 4 4 (0, 2) (0, 2) (1, 0) (1, 0) y
 4.3.58: 2 6 4 4 (0, 6) (0, 6) ( 5, 0) (5, 0) 2 4 y
 4.3.59: 4 ( ) ( ) (2, 0) (2, 0) 3 3 2 2 0, 0, y
 4.3.60: 8 84 8 8 0, 7 2 ( ) 0, 7 2 ( ) ( 7, 0) (7, 0) y
 4.3.61: Vertices: foci:
 4.3.62: Vertices: foci:
 4.3.63: Foci: major axis of length 14
 4.3.64: Foci: major axis of length 10
 4.3.65: Vertices: passes through the point
 4.3.66: Vertical major axis; passes through the points and
 4.3.67: ARCHITECTURE A fireplace arch is to be constructed in the shape of ...
 4.3.68: ARCHITECTURE A semielliptical arch over a tunnel for a oneway road...
 4.3.69: ARCHITECTURE Repeat Exercise 68 for a semielliptical arch with a ma...
 4.3.70: GEOMETRY A line segment through a focus of an ellipse with endpoint...
 4.3.71: x2 4 y 2 1 1
 4.3.72: x2 9 y 2 16 1
 4.3.73: 9x 2 15 2 4y 2 36
 4.3.74: 5x 2 3y 9x 2 15 2
 4.3.75: x 1 2 y 2 1 5
 4.3.76: x 2 9 y 2 16 x 1 2
 4.3.77: y 2 1 x2 4 1 x
 4.3.78: y 2 9 x2 1 1 y
 4.3.79: 2 49 x2 196 1 y
 4.3.80: x2 36 y 2 4 1 y
 4.3.81: 4y 2 36 2 x2 1 x
 4.3.82: 4y2 9x 4y 2 36 2
 4.3.83: 36y2 1 100x2 1 4
 4.3.84: 144x2 1 169 y2 1 1
 4.3.85: Vertices: foci:
 4.3.86: Vertices: foci:
 4.3.87: Vertices: asymptotes:
 4.3.88: Vertices: asymptotes:
 4.3.89: Foci: asymptotes:
 4.3.90: Foci: asymptotes:
 4.3.91: Vertices: passes through the point
 4.3.92: Vertices: passes through the point
 4.3.93: ART A sculpture has a hyperbolic cross section (see figure). (a) Wr...
 4.3.94: OPTICS A hyperbolic mirror (used in some telescopes) has the proper...
 4.3.95: AERONAUTICS When an airplane travels faster than the speed of sound...
 4.3.96: NAVIGATION Long distance radio navigation for aircraft and ships us...
 4.3.97: The equation represents a circle.
 4.3.98: The major axis of the ellipse is vertical.
 4.3.99: It is possible for a parabola to intersect its directrix.
 4.3.100: If the vertex and focus of a parabola are on a horizontal line, the...
 4.3.101: Consider the ellipse (a) The area of the ellipse is given by Write ...
 4.3.102: CAPSTONE Identify the conic. Explain your reasoning.
 4.3.103: THINK ABOUT IT How can you tell if an ellipse is a circle from the ...
 4.3.104: THINK ABOUT IT Is the graph of an ellipse? Explain.
 4.3.105: THINK ABOUT IT The graph of is a degenerate conic. Sketch this grap...
 4.3.106: THINK ABOUT IT Which part of the graph of the ellipse is represente...
 4.3.107: WRITING At the beginning of this section, you learned that each typ...
 4.3.108: WRITING Write a paragraph discussing the changes in the shape and o...
 4.3.109: Use the definition of an ellipse to derive the standard form of the...
 4.3.110: Use the definition of a hyperbola to derive the standard form of th...
 4.3.111: An ellipse can be drawn using two thumbtacks placed at the foci of ...
Solutions for Chapter 4.3: Conics
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 4.3: Conics
Get Full SolutionsSince 111 problems in chapter 4.3: Conics have been answered, more than 51198 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Chapter 4.3: Conics includes 111 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474.

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.)

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

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).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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

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