Answer: The eccentricity e of a hyperbola is defined as

The distance d from P1 = 13, -42 to P2 = 1 -2, 12 is d = . (p. 5)

To complete the square of x2 + 5x, add . (pp. A28A29)

To complete the square of x2 + 5x, add . (pp. A28A29)

True or False The equation y2 = 9 + x2 is symmetric with

To graph y = 1x - 523 - 4, shift the graph of y = x3 to the (left/right) unit(s) and then (up/down) unit(s). (pp. 104106)

Find the vertical asymptotes, if any, and the horizontal or oblique asymptote, if any, of y = x2 - 9 x2 - 4 . (pp. 220221)

A(n) is the collection of points in the plane the difference of whose distances from two fixed points is a constant.

For a hyperbola, the foci lie on a line called the

The equation of the hyperbola is of the form (a) 1x - h22 a2 - 1y - k22 b2 = 1 (b) 1y - k22 a2 - 1x - h22 b2 = 1

If the center of the hyperbola is (2, 1) and a = 3, then the coordinates of the vertices are and .

If the center of the hyperbola is (2, 1) and c = 5, then the coordinates of the foci are and .

In a hyperbola, if a = 3 and c = 5, then b .

For the hyperbola x2 4 - y2 9 = 1, the value of a is , the value of b is , and the transverse axis is the -axis.

For the hyperbola y2 16 - x2 81 = 1, the asymptotes are and .

In Problems 1518, the graph of a hyperbola is given. Match each graph to its equation. (A) x2 4 - y2 = 1 (B) x2 - y2 4 = 1 (C) y2 4 - x2 = 1 (D) y2 - x2 4 = 1

In Problems 1518, the graph of a hyperbola is given. Match each graph to its equation. (A) x2 4 - y2 = 1 (B) x2 - y2 4 = 1 (C) y2 4 - x2 = 1 (D) y2 - x2 4 = 2

In Problems 1518, the graph of a hyperbola is given. Match each graph to its equation. (A) x2 4 - y2 = 1 (B) x2 - y2 4 = 1 (C) y2 4 - x2 = 1 (D) y2 - x2 4 = 3

In Problems 1518, the graph of a hyperbola is given. Match each graph to its equation. (A) x2 4 - y2 = 1 (B) x2 - y2 4 = 1 (C) y2 4 - x2 = 1 (D) y2 - x2 4 = 4

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Center at 10, 02; focus at 13, 02; vertex at 11, 02 2

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Center at 10, 02; focus at 10, 52; vertex at 10, 32

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Center at 10, 02; focus at 10, -62; vertex at 10, 42

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Center at 10, 02; focus at 1 -3, 02; vertex at 12, 02

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Foci at 1 -5, 02 and 15, 02; vertex at 13, 02

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Focus at 10, 62; vertices at 10, -22 and 10, 22

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Vertices at 10, -62 and 10, 62; asymptote the line y = 2x

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Vertices at 1 -4, 02 and 14, 02; asymptote the line y = 2x

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Foci at 1 -4, 02 and 14, 02; asymptote the line y = -x

In Problems 1928, find an equation for the hyperbola described. Graph the equation by hand.Foci at 1 -4, 02 and 14, 02; asymptote the line y = -x

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 2936, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 3740, write an equation for each hyperbola.

In Problems 3740, write an equation for each hyperbola.

In Problems 3740, write an equation for each hyperbola.

In Problems 3740, write an equation for each hyperbola.

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Center at 14, -12; focus at 17, -12; vertex at 16, -12

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Center at 1 -3, 12; focus at 1 -3, 62; vertex at 1 -3, 42

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Center at 1 -3, -42; focus at 1 -3, -82; vertex at 1 -3, -22

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Center at 11, 42; focus at 1 -2, 42; vertex at 10, 42

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Foci at 13, 72 and 17, 72; vertex at 16, 72

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Focus at 1 -4, 02 vertices at 1 -4, 42 and 1 -4, 22

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Vertices at 1 -1, -12 and 13, -12; asymptote the line y + 1 = 3 2 1x - 12

In Problems 41 48, find an equation for the hyperbola described. Graph the equation by hand.Vertices at 11, -32 and 11, 12; asymptote the line y + 1 = 3 2 1x - 12

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 49 62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility.

In Problems 63 66, graph each function by hand. Be sure to label any intercepts. [Hint: Notice that each function is half a hyperbola.]

In Problems 63 66, graph each function by hand. Be sure to label any intercepts. [Hint: Notice that each function is half a hyperbola.]

In Problems 63 66, graph each function by hand. Be sure to label any intercepts. [Hint: Notice that each function is half a hyperbola.]

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

In Problems 6774, analyze each conic.

Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time, the first person standing at point A hears the burst. One second later, the second person standing at point B hears the burst. If the person at point B is due west of the person at point A and if the display is known to occur due north of the person at point A, where did the fireworks display occur?

Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time, the first person standing at point A hears the burst. One second later, the second person standing at point B hears the burst. If the person at point B is due west of the person at point A and if the display is known to occur due north of the person at point A, where did the fireworks display occur?

Some nuclear power plants utilize natural draft cooling towers in the shape of a hyperboloid, a solid obtained by rotating a hyperbola about its conjugate axis. Suppose that such a cooling tower has a base diameter of 400 feet and the diameter at its narrowest point, 360 feet above the ground, is 200 feet. If the diameter at the top of the tower is 300 feet, how tall is the tower?

Some nuclear power plants utilize natural draft cooling towers in the shape of a hyperboloid, a solid obtained by rotating a hyperbola about its conjugate axis. Suppose that such a cooling tower has a base diameter of 400 feet and the diameter at its narrowest point, 360 feet above the ground, is 200 feet. If the diameter at the top of the tower is 300 feet, how tall is the tower?

In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil 0.00004 cm thick. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The figure shows a diagram from Rutherfords paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola.

Hyperbolas have interesting reflective properties that make them useful for lenses and mirrors. For example, if a ray of light strikes a convex hyperbolic mirror on a line that would (theoretically) pass through its rear focus, it is reflected through the front focus. This property, and that of the parabola, were used to develop the Cassegrain telescope in 1672. The focus of the parabolic mirror and the rear focus of the hyperbolic mirror are the same point. The rays are collected by the parabolic mirror, reflected toward the (common) focus, and thus are reflected by the hyperbolic mirror through the opening to its front focus, where the eyepiece is located. If the equation of the hyperbola is y2 9 - x2 16 = 1 and the focal length (distance from the vertex to the focus) of the parabola is 6, find the equation of the parabola

The eccentricity e of a hyperbola is defined as the number c a , where a is the distance of a vertex from the center and c is the distance of a focus from the center. Because c 7 a, it follows that e 7 1. Describe the general shape of a hyperbola whose eccentricity is close to 1. What is the shape if e is very large?

A hyperbola for which a = b is called an equilateral hyperbola. Find the eccentricity e of an equilateral hyperbola. [Note: The eccentricity of a hyperbola is defined in Problem 81.

Two hyperbolas that have the same set of asymptotes are called conjugate. Show that the hyperbolas x2 4 - y2 = 1 and y2 - x2 4 = 1 are conjugate. Graph each hyperbola on the same set of coordinate axes.

Prove that the hyperbola y2 a2 - x2 b2 = 1 has the two oblique asymptotes y = a b x and y = - a b x

Show that the graph of an equation of the form Ax2 + Cy2 + F = 0 A 0, C 0, F 0 where A and C are of opposite sign, is a hyperbola with center at 10, 02.

Show that the graph of an equation of the form Ax2 + Cy2 + Dx + Ey + F = 0 A 0, C 0 where A and C are of opposite sign, (a) is a hyperbola if D2 4A + E2 4C - F 0. (b) is two intersecting lines if D2 4A + E2 4C - F = 0. A