 4.3.1: Fill in the blanks. A ________ is the intersection of a plane and a...
 4.3.2: Fill in the blanks. The equation is the standard form of the equati...
 4.3.3: Fill in the blanks. A ________ is the set of all points in a plane ...
 4.3.4: Fill in the blanks. The ________ of a parabola is the midpoint betw...
 4.3.5: Fill in the blanks. The line that passes through the focus and the ...
 4.3.6: Fill in the blanks. An ________ is the set of all points in a plane...
 4.3.7: Fill in the blanks. The chord joining the vertices of an ellipse is...
 4.3.8: Fill in the blanks. The chord perpendicular to the major axis at th...
 4.3.9: Fill in the blanks. A ________ is the set of all points in a plane,...
 4.3.10: Fill in the blanks. The line segment connecting the vertices of a h...
 4.3.11: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.12: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.13: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.14: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.15: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.16: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.17: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.18: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.19: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.20: In Exercises 1120, match the equation with its graph. If the graph ...
 4.3.21: In Exercises 2126, find the vertex and focus of the parabola and sk...
 4.3.22: In Exercises 2126, find the vertex and focus of the parabola and sk...
 4.3.23: In Exercises 2126, find the vertex and focus of the parabola and sk...
 4.3.24: In Exercises 2126, find the vertex and focus of the parabola and sk...
 4.3.25: In Exercises 2126, find the vertex and focus of the parabola and sk...
 4.3.26: In Exercises 2126, find the vertex and focus of the parabola and sk...
 4.3.27: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.28: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.29: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.30: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.31: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.32: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.33: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.34: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.35: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.36: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.37: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.38: In Exercises 2738, find the standard form of the equation of the pa...
 4.3.39: In Exercises 39 42, find the standard form of the equation of the p...
 4.3.40: In Exercises 39 42, find the standard form of the equation of the p...
 4.3.41: In Exercises 39 42, find the standard form of the equation of the p...
 4.3.42: In Exercises 39 42, find the standard form of the equation of the p...
 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: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.48: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.49: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.50: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.51: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.52: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.53: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.54: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.55: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.56: In Exercises 4756, find the center and vertices of the ellipse and ...
 4.3.57: In Exercises 5766, find the standard form of the equation of the el...
 4.3.58: In Exercises 5766, find the standard form of the equation of the el...
 4.3.59: In Exercises 5766, find the standard form of the equation of the el...
 4.3.60: In Exercises 5766, find the standard form of the equation of the el...
 4.3.61: In Exercises 5766, find the standard form of the equation of the el...
 4.3.62: In Exercises 5766, find the standard form of the equation of the el...
 4.3.63: In Exercises 5766, find the standard form of the equation of the el...
 4.3.64: In Exercises 5766, find the standard form of the equation of the el...
 4.3.65: In Exercises 5766, find the standard form of the equation of the el...
 4.3.66: In Exercises 5766, find the standard form of the equation of the el...
 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: In Exercises 7174, sketch the graph of the ellipse, using the later...
 4.3.72: In Exercises 7174, sketch the graph of the ellipse, using the later...
 4.3.73: In Exercises 7174, sketch the graph of the ellipse, using the later...
 4.3.74: In Exercises 7174, sketch the graph of the ellipse, using the later...
 4.3.75: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.76: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.77: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.78: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.79: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.80: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.81: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.82: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.83: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.84: In Exercises 7584, find the center and vertices of the hyperbola an...
 4.3.85: In Exercises 8592, find the standard form of the equation of the hy...
 4.3.86: In Exercises 8592, find the standard form of the equation of the hy...
 4.3.87: In Exercises 8592, find the standard form of the equation of the hy...
 4.3.88: In Exercises 8592, find the standard form of the equation of the hy...
 4.3.89: In Exercises 8592, find the standard form of the equation of the hy...
 4.3.90: In Exercises 8592, find the standard form of the equation of the hy...
 4.3.91: In Exercises 8592, find the standard form of the equation of the hy...
 4.3.92: In Exercises 8592, find the standard form of the equation of the hy...
 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: TRUE OR FALSE? In Exercises 97100, determine whether the statement ...
 4.3.98: TRUE OR FALSE? In Exercises 97100, determine whether the statement ...
 4.3.99: TRUE OR FALSE? In Exercises 97100, determine whether the statement ...
 4.3.100: TRUE OR FALSE? In Exercises 97100, determine whether the statement ...
 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. (a) (b) (c) (d...
 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 College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 4.3: CONICS
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra , edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.3: CONICS includes 111 full stepbystep solutions. Since 111 problems in chapter 4.3: CONICS have been answered, more than 36675 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781439048696.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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.

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

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.

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.

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.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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 .

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.

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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