In Exercises 6770, you may use a graphing calculator to

In Exercises 1 and 2, find the distance between the given points. 1-3 -2 21

In Exercises 1 and 2, find the distance between the given points. 1-3 -4 21

In Exercises 3 and 4, solve for y in terms of x.2 9 + x 2 4 = 1

In Exercises 3 and 4, solve for y in terms of x.x 2 36 + y2 25 = 1 y

In Exercises 58, solve for x algebraically.13x + 12 + 13x - 8 = 10

In Exercises 58, solve for x algebraically.16x + 12 - 14x + 9 = 1

In Exercises 58, solve for x algebraically.26x 2 + 12 + 26x 2 + 1 = 11

In Exercises 58, solve for x algebraically.22x 2 + 8 + 23x 2 + 4 = 8

In Exercises 9 and 10, find exact solutions by completing the square.2x2 - 6x - 3 = 0

In Exercises 9 and 10, find exact solutions by completing the square.2 2x + 4x - 5 = 0

In Exercises 16, find the vertices and foci of the ellipse.x 2 16 + y2 7 = 1

In Exercises 16, find the vertices and foci of the ellipse.y2 25 + x 2 21 = 1

In Exercises 16, find the vertices and foci of the ellipse.y2 36 + x 2 27 = 1

In Exercises 16, find the vertices and foci of the ellipse.x 2 11 + y2 7 = 1

In Exercises 16, find the vertices and foci of the ellipse.3x 2 + 4y2 = 12

In Exercises 16, find the vertices and foci of the ellipse.9x 2 + 4y2 = 36

In Exercises 710, match the graph with its equation, given that the tick marks on all axes are 1 unit apart.x 2 25 + y2 16 = 1

In Exercises 710, match the graph with its equation, given that the tick marks on all axes are 1 unit apart.y2 36 + x 2 9 = 1

In Exercises 710, match the graph with its equation, given that the tick marks on all axes are 1 unit apart.1y - 222 16 + 1x + 322 4 = 1

In Exercises 710, match the graph with its equation, given that the tick marks on all axes are 1 unit apart.1x - 122 11 + 1y + 222 = 1

In Exercises 1116, sketch the graph of the ellipse by hand.x 2 64 + y2 36 = 1

In Exercises 1116, sketch the graph of the ellipse by hand.x 2 81 + y2 25 = 1

In Exercises 1116, sketch the graph of the ellipse by hand.y2 9 + x 2 4 = 1

In Exercises 1116, sketch the graph of the ellipse by hand.y2 49 + x 2 25 = 1

In Exercises 1116, sketch the graph of the ellipse by hand.1x + 322 16 + 1y - 122 4 = 1

In Exercises 1116, sketch the graph of the ellipse by hand.x - 122 2 + 1y + 322 4 = 1

In Exercises 1720, graph the ellipse using a function grapher.x 2 36 + y2 16 = 1

In Exercises 1720, graph the ellipse using a function grapher.y2 64 + x 2 16 = 1 x

In Exercises 1720, graph the ellipse using a function grapher.1x + 222 5 + 21y - 122 = 1

In Exercises 1720, graph the ellipse using a function grapher.1x - 422 16 + 161y + 422 = 8

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis length 6 on y-axis, minor axis length 4

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis length 14 on x-axis, minor axis length 10

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Foci 2, 0 , major axis length 10

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Foci 0, 3 , major axis length 10

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Endpoints of axes are 4, 0 and 0, 5

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Endpoints of axes are 7, 0 and 0, 4

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis endpoints 0, 6 , minor axis length 8

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis endpoints 5, 0 , minor axis length 4

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Minor axis endpoints 0, 4 , major axis length 10

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Minor axis endpoints 12, 0 , major axis length 26

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis endpoints 1, and 1, 8 , minor axis length 8

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis endpoints , and , 7 , minor axis length 4

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Foci and , major axis endpoints and 16, -42

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Foci and , major axis endpoints and 1-2, 72

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis endpoints and , minor axis length 6

In Exercises 2136, find an equation in standard form for the ellipse that satisfies the given conditions.Major axis endpoints , 2 and 3, 2 , minor axis length 6

In Exercises 3740, find the center, vertices, and foci of the ellipse.1x + 122 25 + 1y - 222 16 = 1

In Exercises 3740, find the center, vertices, and foci of the ellipse.1x - 322 11 + 1y - 522 7 = 1

In Exercises 3740, find the center, vertices, and foci of the ellipse.1y + 322 81 + 1x - 722 64 = 1

In Exercises 3740, find the center, vertices, and foci of the ellipse.1y - 122 25 + 1x + 222 16 = 1

In Exercises 4144, graph the ellipse using a parametric grapher.y2 25 + x 2 4 = 1

In Exercises 4144, graph the ellipse using a parametric grapher.x 2 30 + y2 20 = 1

In Exercises 41–44, graph the ellipse using a parametric grapher. \(\frac{(x+3)^{2}}{12}+\frac{(y-6)^{2}}{5}=1\)

In Exercises 41–44, graph the ellipse using a parametric grapher. \(\frac{(y+1)^{2}}{15}+\frac{(x-2)^{2}}{6}=1\)

In Exercises 45– 48, prove that the graph of the equation is an ellipse, and find its vertices, foci, and eccentricity. \(9 x^{2}+4 y^{2}-18 x+8 y-23=0\)

In Exercises 45– 48, prove that the graph of the equation is an ellipse, and find its vertices, foci, and eccentricity. \(3 x^{2}+5 y^{2}-12 x+30 y+42=0\)

In Exercises 45– 48, prove that the graph of the equation is an ellipse, and find its vertices, foci, and eccentricity. \(9 x^{2}+16 y^{2}+54 x-32 y-47=0\)

In Exercises 45– 48, prove that the graph of the equation is an ellipse, and find its vertices, foci, and eccentricity. \(4 x^{2}+y^{2}-32 x+16 y+124=0\)

In Exercises 49 and 50, write an equation for the ellipse.

In Exercises 49 and 50, write an equation for the ellipse.

Writing to Learn Prove that an equation for the ellipse with center (0, 0), foci \((0,\ \pm c)\) and semimajor axis \(a>c \geq 0\) is \(y^{2} / a^{2}+x^{2} / b^{2}=1\), where \(b^{2}=a^{2}-c^{2}\). [Hint: Refer to derivation at the beginning of the section.]

Writing to Learn Dancing Among the Stars Using the data in Table 8.1, prove that the object with the most eccentric orbit sometimes is closer to the Sun than the planet with the least eccentric orbit.

The Moon’s Orbit The Moon’s apogee (farthest distance from Earth) is 252,710 miles, and perigee (closest distance to Earth) is 221,463 miles. Assuming the Moon’s orbit of Earth is elliptical with Earth at one focus, calculate and interpret a, b, c, and e.

Hot Mercury Given that the diameter of the Sun is about 1.392 Gm, how close does Mercury get to the Sun’s surface?

Saturn Find the perihelion and aphelion distances of Saturn.

Venus and Mars Write equations for the orbits of Venus and Mars in the form \(x^{2} / a^{2}+y^{2} / b^{2}=1\).

Sungrazers One comet group, known as the sungrazers, passes within a Sun’s diameter (1.392 Gm) of the solar surface. What can you conclude about a - c for orbits of the sungrazers?

Halley’s Comet The orbit of Halley’s comet is 36.18 AU long and 9.12 AU wide. What is its eccentricity?

Lithotripter For an ellipse that generates the ellipsoid of a lithotripter, the major axis has endpoints (-8, 0) and (8, 0). One endpoint of the minor axis is (0, 3.5). Find the coordinates of the foci.

Lithotripter (Refer to Figure 8.21.) A lithotripter’s shape is formed by rotating the portion of an ellipse below its minor axis about its major axis. If the length of the major axis is 26 in. and the length of the minor axis is 10 in., where should the shock-wave source and the patient be placed for maximum effect?

Group Activities In Exercises 61 and 62, solve the system of equations algebraically and support your answer graphically. \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) \(x^{2}+y^{2}=4\)

Group Activities In Exercises 61 and 62, solve the system of equations algebraically and support your answer graphically. \(\frac{x^{2}}{9}+y^{2}=1\) \(x-3 y=-3\)

Group Activity Consider the system of equations \(x^{2}+4 y^{2}=4\) \(y=2 x^{2}-3\) (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.

Writing to Learn Look up the adjective eccentric in a dictionary and read its various definitions. Notice that the word is derived from ex-centric, meaning “out-of-center” or “off-center.” Explain how this is related to the word’s everyday meanings as well as its mathematical meaning for ellipses.

True or False The distance from a focus of an ellipse to the closer vertex is a(1 + e), where a is the semimajor axis and e is the eccentricity. Justify your answer.

True or False The distance from a focus of an ellipse to either endpoint of the minor axis is half the length of the major axis. Justify your answer.

In Exercises 67–70, 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) (2, 0). (C) \((\sqrt{3},\ 0)\) (D) \((\sqrt{2},\ 0)\) (E) (1, 0)

In Exercises 67–70, you may use a graphing calculator to solve the problem. Multiple Choice The focal axis of \(\frac{(x-2)^{2}}{25}+\frac{(y-3)^{2}}{16}=1\) is (A) y = 1. (B) y = 2. (C) y = 3. (D) y = 4. (E) y = 5.

In Exercises 67–70, you may use a graphing calculator to solve the problem. Multiple Choice The center of \(9 x^{2}+4 y^{2}-72 x-24 y+144=0\) is (A) (4, 0). (B) (4, 3). (C) (4, 4). (D) (4, 5). (E) (4, 6).

In Exercises 67–70, you may use a graphing calculator to solve the problem. Multiple Choice The perimeter of a triangle with one vertex on the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) and the other two vertices at the foci of the ellipse would be (A) a + b. (B) 2a + 2b. (C) 2a + 2c. (D) 2b + 2c. (E) a + b + c.

Area and Perimeter The area of an ellipse is \(A=\pi a b\), but the perimeter cannot be expressed so simply: \(P \approx \pi(a+b)\left(3-\frac{\sqrt{(3 a+b)(a+3 b)}}{a+b}\right)\) (a) Prove that, when a = b = r, these become the familiar formulas for the area and perimeter (circumference) of a circle. (b) Find a pair of ellipses such that the one with greater area has smaller perimeter.

Writing to Learn Kepler’s Laws We have encountered Kepler’s first and third laws (pages 179, 595). Using a library or the Internet, (a) Read about Kepler’s life, and write in your own words how he came to discover his three laws of planetary motion. (b) What is Kepler’s Second Law? Explain it with both pictures and words.

Pendulum Velocity vs. Position As a pendulum swings toward and away from a motion detector, its distance (in meters) from the detector is given by the position function x(t) = 3 + cos (2t - 5), where t represents time (in seconds). The velocity (in m/sec) of the pendulum is given by y(t) = -2 sin (2t - 5). (a) Using parametric mode on your grapher, plot the (x, y) relation for velocity versus position for \(0 \leq t \leq 2 \pi\). (b) Write the equation of the resulting conic in standard form, in terms of x and y, and eliminating the parameter t.

Pendulum Velocity vs. Position A pendulum that swings toward and away from a motion detector has a distance (in feet) from the detector of \(x(t)=5+3 \sin (\pi t+\pi / 2)\) and a velocity (in ft/sec) of \(y(t)=3 \pi \cos (\pi t+\pi / 2)\) where t represents time (in seconds). (a) Prove that the plot of velocity versus position (distance) is an ellipse. (b) Writing to Learn Describe the motion of the pendulum.

Prove that a nondegenerate graph of the equation \(A x^{2}+C y^{2}+D x+E y+F=0\) is an ellipse if AC > 0.

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 ellipse. Describe the graph. How is it like a full-fledged ellipse, and how is it different?