Precalculus: Graphical, Numerical, Algebraic 8th Edition solutions

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1 and 2, find the distance between the given points. (4, -3) and (-7, -8)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1 and 2, find the distance between the given points. (a, -3) and (b, c)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3 and 4, solve for y in terms of x. \(\frac{y^{2}}{16}-\frac{x^{2}}{9}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3 and 4, solve for y in terms of x. \(\frac{x^{2}}{36}-\frac{y^{2}}{4}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, solve for x. \(\sqrt{3 x+12}-\sqrt{3 x-8}=10\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, solve for x. \(\sqrt{4 x+12}-\sqrt{x+8}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, solve for x. \(\sqrt{6 x^{2}+12}-\sqrt{6 x^{2}+1}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, solve for x. \(\sqrt{2 x^{2}+12}-\sqrt{3 x^{2}+4}=-8\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 9 and 10, solve the system of equations. \(c-a=2\) and \(c^{2}-a^{2}=16 a / 3\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 9 and 10, solve the system of equations. \(c-a=1\) and \(c^{2}-a^{2}=25 a / 12\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find the vertices and foci of the hyperbola. \(\frac{x^{2}}{16}-\frac{y^{2}}{7}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find the vertices and foci of the hyperbola. \(\frac{y^{2}}{25}-\frac{x^{2}}{21}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find the vertices and foci of the hyperbola. \(\frac{y^{2}}{36}-\frac{x^{2}}{13}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find the vertices and foci of the hyperbola. \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find the vertices and foci of the hyperbola. \(3 x^{2}-4 y^{2}=12\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find the vertices and foci of the hyperbola. \(9 x^{2}-4 y^{2}=36\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, match the graph with its equation. \(\frac{x^{2}}{25}-\frac{y^{2}}{16}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, match the graph with its equation. \(\frac{y^{2}}{4}-\frac{x^{2}}{9}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, match the graph with its equation. \(\frac{(y-2)^{2}}{4}-\frac{(x+3)^{2}}{16}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, match the graph with its equation. \(\frac{(x-2)^{2}}{9}-(y+1)^{2}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1116, sketch the graph of the hyperbola by hand. \(\frac{x^{2}}{49}-\frac{y^{2}}{25}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1116, sketch the graph of the hyperbola by hand. \(\frac{y^{2}}{64}-\frac{x^{2}}{25}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1116, sketch the graph of the hyperbola by hand. \(\frac{y^{2}}{25}-\frac{x^{2}}{16}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1116, sketch the graph of the hyperbola by hand. \(\frac{x^{2}}{169}-\frac{y^{2}}{144}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1116, sketch the graph of the hyperbola by hand. \(\frac{(x+3)^{2}}{16}-\frac{(y-1)^{2}}{4}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1116, sketch the graph of the hyperbola by hand. \(\frac{(x-1)^{2}}{2}-\frac{(y+3)^{2}}{4}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1722, graph the hyperbola using a function grapher. \(\frac{x^{2}}{36}-\frac{y^{2}}{16}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1722, graph the hyperbola using a function grapher. \(\frac{y^{2}}{64}-\frac{x^{2}}{16}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1722, graph the hyperbola using a function grapher. \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1722, graph the hyperbola using a function grapher. \(\frac{y^{2}}{16}-\frac{x^{2}}{9}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1722, graph the hyperbola using a function grapher. \(\frac{x^{2}}{4}-\frac{(y-3)^{2}}{5}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1722, graph the hyperbola using a function grapher. \(\frac{(y-3)^{2}}{9}-\frac{(x+2)^{2}}{4}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Foci \((\pm3,\ 0)\), transverse axis length 4

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Foci \((0,\ \pm3)\), transverse axis length 4

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Foci \((0,\ \pm15)\), transverse axis length 8

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Foci \((\pm5,\ 0)\), transverse axis length 3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Center at (0, 0), a = 5, e = 2, horizontal focal axis

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Center at (0, 0), a = 4, e = 3/2, vertical focal axis

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Center at (0, 0), b = 5, e = 13/12, vertical focal axis

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Center at (0, 0), c = 6, e = 2, horizontal focal axis

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Transverse axis endpoints (2, 3) and (2, -1), conjugate axis length 6

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Transverse axis endpoints (5, 3) and (-7, 3), conjugate axis length 10

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Transverse axis endpoints (-1, 3) and (5, 3), slope of one asymptote 4/3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Transverse axis endpoints (-2, -2) and (-2, 7), slope of one asymptote 4/3

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Foci (-4, 2) and (2, 2), transverse axis endpoints (-3, 2) and (1, 2)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Foci (-3, -11) and (-3, 0), transverse axis endpoints (-3, -9) and (-3, -2)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Center at (-3, 6), a = 5, e = 2, vertical focal axis

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 2338, find an equation in standard form for the hyperbola that satisfies the given conditions. Center at (1, -4), c = 6, e = 2, horizontal focal axis

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 39 42, find the center, vertices, and the foci of the hyperbola. \(\frac{(x+1)^{2}}{144}-\frac{(y-2)^{2}}{25}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 39 42, find the center, vertices, and the foci of the hyperbola. \(\frac{(x+4)^{2}}{12}-\frac{(y+6)^{2}}{13}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 39 42, find the center, vertices, and the foci of the hyperbola. \(\frac{(y+3)^{2}}{64}-\frac{(x-2)^{2}}{81}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 39 42, find the center, vertices, and the foci of the hyperbola. \(\frac{(y-1)^{2}}{25}-\frac{(x+5)^{2}}{11}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 43 46, graph the hyperbola using a parametric grapher in Dot graphing mode. \(\frac{y^{2}}{25}-\frac{x^{2}}{4}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 43 46, graph the hyperbola using a parametric grapher in Dot graphing mode. \(\frac{x^{2}}{30}-\frac{y^{2}}{20}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 43 46, graph the hyperbola using a parametric grapher in Dot graphing mode. \(\frac{(x+3)^{2}}{12}-\frac{(y-6)^{2}}{5}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 43 46, graph the hyperbola using a parametric grapher in Dot graphing mode. \(\frac{(y+1)^{2}}{15}-\frac{(x-2)^{2}}{6}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4750, graph the hyperbola, and find its vertices, foci, and eccentricity. \(4(y-1)^{2}-9(x-3)^{2}=36\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4750, graph the hyperbola, and find its vertices, foci, and eccentricity. \(4(x-2)^{2}-9(y+4)^{2}=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4750, graph the hyperbola, and find its vertices, foci, and eccentricity. \(9 x^{2}-4 y^{2}-36 x+8 y-4=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4750, graph the hyperbola, and find its vertices, foci, and eccentricity. \(25 y^{2}-9 x^{2}-50 y-54 x-281=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 51 and 52, write an equation for the hyperbola.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 51 and 52, write an equation for the hyperbola.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Prove that an equation for the hyperbola with center (0, 0), foci \((0,\ \pm c)\), and semitransverse axis a is \(y^{2} / a^{2}-x^{2} / b^{2}=1\), where c > a > 0 and \(b^{2}=c^{2}-a^{2}\). [Hint: Refer to derivation at the beginning of the section.]

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Degenerate Hyperbolas Graph the degenerate hyperbola. (a) \(\frac{x^{2}}{4}-y^{2}=0\) (b) \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=0\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Rogue Comet A comet following a hyperbolic path about the Sun has a perihelion of 120 Gm. When the line from the comet to the Sun is perpendicular to the focal axis of the orbit, the comet is 250 Gm from the Sun. Calculate a, b, c, and e. What are the coordinates of the center of the Sun if the cente

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Rogue Comet A comet following a hyperbolic path about the Sun has a perihelion of 140 Gm. When the line from the comet to the Sun is perpendicular to the focal axis of the orbit, the comet is 405 Gm from the Sun. Calculate a, b, c, and e. What are the coordinates of the center of the Sun if the cente

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Long-Range Navigation Three LORAN radio transmitters are positioned as shown in the figure, with R due north of O and Q due east of O. The cruise ship Princess Ann receives simultaneous signals from the three transmitters. The signal from O arrives \(323.27\ \mu\mathrm{\sec}\) after the signal from R

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Gun Location Observers are located at positions A, B, and C with A due north of B. A cannon is located somewhere in the first quadrant as illustrated in the figure. A hears the sound of the cannon 2 sec before B, and C hears the sound 4 sec before B. Determine the bearing and distance of the cannon f

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activities In Exercises 59 and 60, solve the system of equations algebraically and support your answer graphically. \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=1\) \(x-\frac{2 \sqrt{3}}{3} y=-2\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activities In Exercises 59 and 60, solve the system of equations algebraically and support your answer graphically. \(\frac{x^{2}}{4}-y^{2}=1\) \(x^{2}+y^{2}=9\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activity Consider the system of equations \(\frac{x^{2}}{4}-\frac{y^{2}}{25}=1\) \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\). (a) Solve the system graphically. (b) If you have access to a grapher that also does symbolic algebra, use it to find the exact solutions to the system.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Escape of the Unbound When NASA launches a space probe, the probe reaches a speed sufficient for it to become unbound from Earth and escape along a hyperbolic trajectory. Look up escape speed in an astronomy textbook or on the Internet, and write a paragraph in your own words about w

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The distance from a focus of a hyperbola to the closer vertex is a(e - 1) where a is the semitransverse axis and e is the eccentricity. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False Unlike that for an ellipse, the Pythagorean relation for a hyperbola is the usual \(a^{2}+b^{2}=c^{2}\). Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6568, you may use a graphing calculator to solve the problem. Multiple Choice One focus of \(x^{2}-4 y^{2}=4\) is (A) (4, 0). (B) \((\sqrt{5},\ 0)\). (C) (2, 0) (D) \((\sqrt{3},\ 0)\). (E) (1, 0)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6568, you may use a graphing calculator to solve the problem. Multiple Choice The focal axis of \(\frac{(x+5)^{2}}{9}-\frac{(y-6)^{2}}{16}=1\) is (A) y = 2. (B) y = 3. (C) y = 4. (D) y = 5. (E) y = 6.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6568, you may use a graphing calculator to solve the problem. Multiple Choice The center of \(4 x^{2}-12 y^{2}-16 x-72 y-44=0\) is (A) (2, -2). (B) (2, -3). (C) (2, -4) (D) (2, -5). (E) (2, -6)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6568, you may use a graphing calculator to solve the problem. Multiple Choice The slopes of the asymptotes of the hyperbola \(\frac{x^{2}}{4}-\frac{y^{2}}{3}=1\) are (A) \(\pm 1\). (B) \(\pm 3 / 2\). (C) \(\pm \sqrt{3} / 2\) (D) \(\pm 2 / 3\). (E) \(\pm 4 / 3\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Constructing Points of a Hyperbola Use a geometry software package, such as Cabri Geometry \(I I^{\mathrm{TM}}\), The Geometers Sketchpad, or a similar application on a handheld device, to carry out the following construction. (a) Start by placing the coordinate axes in the construction window.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Cassegrain Telescope A Cassegrain telescope as described in the section has the dimensions shown in the figure. Find the standard form for the equation of the hyperbola centered at the origin with the x-axis as the focal axis.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Prove that a nondegenerate graph of the equation \(A x^{2}+C y^{2}+D x+E y+F=0\) is a hyperbola if AC < 0.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn The graph of the equation \(\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=0\) is considered to be a degenerate hyperbola. Describe the graph. How is it like a full-fledged hyperbola, and how is it different?

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Conjugate Hyperbolas The hyperbolas \(\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1\) and \(\frac{(y-k)^{2}}{b^{2}}-\frac{(x-h)^{2}}{a^{2}}=1\) obtained by switching the order of subtraction in their standard equations are conjugate hyperbolas. Prove that these hyperbolas have the same as

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Focal Width of a Hyperbola Prove that for the hy-perbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), if x = c, then \(y=\pm b^{2} / a\). Why is it reasonable to define the focal width of such hyperbolas to be \(2 b^{2} / a\)?

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Explain how the standard form equations for the conics are related to \(A x^{2}+B x y+C y^{2}+D x+E y+F=0\).