Stewart 9e Section 10.5: Conic Sections

1 Parabolas - Standard Form · Level 2

Find the vertex, focus, and directrix of the parabola \(x^2 = 6y\) and sketch its graph.

2 Parabolas - Standard Form · Level 2

Find the vertex, focus, and directrix of the parabola \(2y^2 = 5x\) and sketch its graph.

3 Parabolas - Standard Form · Level 2

Find the vertex, focus, and directrix of the parabola \(2x = -y^2\) and sketch its graph.

4 Parabolas - Standard Form · Level 2

Find the vertex, focus, and directrix of the parabola \(3x^2 + 8y = 0\) and sketch its graph.

5 Parabolas - Shifted · Level 2

Find the vertex, focus, and directrix of the parabola \((x + 2)^2 = 8(y - 3)\) and sketch its graph.

6 Parabolas - Shifted · Level 2

Find the vertex, focus, and directrix of the parabola \((y - 2)^2 = 2x + 1\) and sketch its graph.

7 Parabolas - Completing the Square · Level 3

Find the vertex, focus, and directrix of the parabola \(y^2 + 6y + 2x + 1 = 0\) and sketch its graph.

8 Parabolas - Completing the Square · Level 3

Find the vertex, focus, and directrix of the parabola \(2x^2 - 16x - 3y + 38 = 0\) and sketch its graph.

9 Parabolas - From Figure · Level 2

Find an equation of the parabola shown in the figure. Then find the focus and directrix.

10 Parabolas - From Figure · Level 2

Find an equation of the parabola shown in the figure. Then find the focus and directrix.

11 Ellipses - Standard Form · Level 2

Find the vertices and foci of the ellipse \(\dfrac{x^2}{2} + \dfrac{y^2}{4} = 1\) and sketch its graph.

12 Ellipses - Standard Form · Level 2

Find the vertices and foci of the ellipse \(\dfrac{x^2}{36} + \dfrac{y^2}{8} = 1\) and sketch its graph.

13 Ellipses - Standard Form · Level 2

Find the vertices and foci of the ellipse \(x^2 + 9 y^2 = 9\) and sketch its graph.

14 Ellipses - Standard Form · Level 2

Find the vertices and foci of the ellipse \(100 x^2 + 36 y^2 = 225\) and sketch its graph.

15 Ellipses - Completing the Square · Level 3

Find the vertices and foci of the ellipse \(9 x^2 - 18 x + 4 y^2 = 27\) and sketch its graph.

16 Ellipses - Completing the Square · Level 3

Find the vertices and foci of the ellipse \(x^2 + 3 y^2 + 2 x - 12 y + 10 = 0\) and sketch its graph.

17 Ellipses - From Figure · Level 2

Find an equation of the ellipse shown in the figure. Then find its foci.

18 Ellipses - From Figure · Level 2

Find an equation of the ellipse shown in the figure. Then find its foci.

19 Hyperbolas - Standard Form · Level 2

Find the vertices, foci, and asymptotes of the hyperbola \(\dfrac{y^2}{25} - \dfrac{x^2}{9} = 1\) and sketch its graph.

20 Hyperbolas - Standard Form · Level 2

Find the vertices, foci, and asymptotes of the hyperbola \(\dfrac{x^2}{36} - \dfrac{y^2}{64} = 1\) and sketch its graph.

21 Hyperbolas - Standard Form · Level 2

Find the vertices, foci, and asymptotes of the hyperbola \(x^2 - y^2 = 100\) and sketch its graph.

22 Hyperbolas - Standard Form · Level 2

Find the vertices, foci, and asymptotes of the hyperbola \(y^2 - 16 x^2 = 16\) and sketch its graph.

23 Hyperbolas - Completing the Square · Level 3

Find the vertices, foci, and asymptotes of the hyperbola \(x^2 - y^2 + 2 y = 2\) and sketch its graph.

24 Hyperbolas - Completing the Square · Level 3

Find the vertices, foci, and asymptotes of the hyperbola \(9 y^2 - 4 x^2 - 36 y - 8 x = 4\) and sketch its graph.

25 Identify Conic Section · Level 2

Identify the type of conic section whose equation is given and find the vertices and foci: \(4 x^2 = y^2 + 4\).

26 Identify Conic Section · Level 2

Identify the type of conic section whose equation is given and find the vertices and foci: \(4 x^2 = y + 4\).

27 Identify Conic Section · Level 3

Identify the type of conic section whose equation is given and find the vertices and foci: \(x^2 = 4 y - 2 y^2\).

28 Identify Conic Section · Level 3

Identify the type of conic section whose equation is given and find the vertices and foci: \(y^2 - 2 = x^2 - 2 x\).

29 Identify Conic Section · Level 3

Identify the type of conic section whose equation is given and find the vertices and foci: \(3 x^2 - 6 x - 2 y = 1\).

30 Identify Conic Section · Level 3

Identify the type of conic section whose equation is given and find the vertices and foci: \(x^2 - 2 x + 2 y^2 - 8 y + 7 = 0\).

31 Find Equation - Parabola · Level 2

Find an equation for the conic that satisfies the given conditions: Parabola, vertex \((0, 0)\), focus \((1, 0)\).

32 Find Equation - Parabola · Level 2

Find an equation for the conic that satisfies the given conditions: Parabola, focus \((0, 0)\), directrix \(y = 6\).

33 Find Equation - Parabola · Level 2

Find an equation for the conic that satisfies the given conditions: Parabola, focus \((-4, 0)\), directrix \(x = 2\).

34 Find Equation - Parabola · Level 2

Find an equation for the conic that satisfies the given conditions: Parabola, focus \((2, -1)\), vertex \((2, 3)\).

35 Find Equation - Parabola · Level 3

Find an equation for the conic that satisfies the given conditions: Parabola, vertex \((3, -1)\), horizontal axis, passing through \((-15, 2)\).

36 Find Equation - Parabola · Level 3

Find an equation for the conic that satisfies the given conditions: Parabola, vertical axis, passing through \((0, 4)\), \((1, 3)\), and \((-2, -6)\).

37 Find Equation - Ellipse · Level 2

Find an equation for the conic that satisfies the given conditions: Ellipse, foci \((\pm 2, 0)\), vertices \((\pm 5, 0)\).

38 Find Equation - Ellipse · Level 2

Find an equation for the conic that satisfies the given conditions: Ellipse, foci \((0, \pm \sqrt{2})\), vertices \((0, \pm 2)\).

39 Find Equation - Ellipse · Level 2

Find an equation for the conic that satisfies the given conditions: Ellipse, foci \((0, 2), (0, 6)\), vertices \((0, 0), (0, 8)\).

40 Find Equation - Ellipse · Level 3

Find an equation for the conic that satisfies the given conditions: Ellipse, foci \((0, -1), (8, -1)\), vertex \((9, -1)\).

41 Find Equation - Ellipse · Level 3

Find an equation for the conic that satisfies the given conditions: Ellipse, center \((-1, 4)\), vertex \((-1, 0)\), focus \((-1, 6)\).

42 Find Equation - Ellipse · Level 4

Find an equation for the conic that satisfies the given conditions: Ellipse, foci \((\pm 4, 0)\), passing through \((-4, 1.8)\).

43 Find Equation - Hyperbola · Level 2

Find an equation for the conic that satisfies the given conditions: Hyperbola, vertices \((\pm 3, 0)\), foci \((\pm 5, 0)\).

44 Find Equation - Hyperbola · Level 2

Find an equation for the conic that satisfies the given conditions: Hyperbola, vertices \((0, \pm 2)\), foci \((0, \pm 5)\).

45 Find Equation - Hyperbola · Level 3

Find an equation for the conic that satisfies the given conditions: Hyperbola, vertices \((-3, -4), (-3, 6)\), foci \((-3, -7), (-3, 9)\).

46 Find Equation - Hyperbola · Level 3

Find an equation for the conic that satisfies the given conditions: Hyperbola, vertices \((-1, 2), (7, 2)\), foci \((-2, 2), (8, 2)\).

47 Find Equation - Hyperbola · Level 3

Find an equation for the conic that satisfies the given conditions: Hyperbola, vertices \((\pm 3, 0)\), asymptotes \(y = \pm 2 x\).

48 Find Equation - Hyperbola · Level 4

Find an equation for the conic that satisfies the given conditions: Hyperbola, foci \((2, 0), (2, 8)\), asymptotes \(y = 3 + \dfrac{1}{2} x\) and \(y = 5 - \dfrac{1}{2} x\).

49 Applications - Lunar Orbit · Level 4

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 km and apolune altitude 314 km (above the moon). Find an equation of this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus.

50 Applications - Parabolic Reflector · Level 3

A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm. (a) Find an equation of the parabola. (b) Find the diameter of the opening \(|C D|\), 11 cm from the vertex.

51 Applications - LORAN Navigation · Level 5

The LORAN (LOng RAnge Navigation) radio navigation system was widely used until the 1990s when it was superseded by the GPS system. In the LORAN system, two radio stations located at \(A\) and \(B\) transmit simultaneous signals to a ship or an aircraft located at \(P\). The onboard computer converts the time difference in receiving these signals into a distance difference \(|P A| - |P B|\), and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola. Suppose that station \(B\) is located 400 mi due east of station \(A\) on a coastline. A ship received the signal from \(B\) 1200 microseconds (μs) before it received the signal from \(A\). (a) Assuming that radio signals travel at a speed of 980 ft/μs, find an equation of the hyperbola on which the ship lies. (b) If the ship is due north of \(B\), how far off the coastline is the ship?

52 Hyperbola - Derivation · Level 4

Use the definition of a hyperbola to derive Equation 6 for a hyperbola with foci \((\pm c, 0)\) and vertices \((\pm a, 0)\).

53 Hyperbola - Concavity · Level 4

Show that the function defined by the upper branch of the hyperbola \(y^2 / a^2 - x^2 / b^2 = 1\) is concave upward.

54 Find Equation - Rotated Ellipse · Level 4

Find an equation for the ellipse with foci \((1, 1)\) and \((-1, -1)\) and major axis of length 4.

55 Identify Conic - Parameterized · Level 4

Determine the type of curve represented by the equation \(\dfrac{x^2}{k} + \dfrac{y^2}{k - 16} = 1\) in each of the following cases: (a) \(k > 16\), (b) \(0 < k < 16\), (c) \(k < 0\). (d) Show that all the curves in parts (a) and (b) have the same foci, no matter what the value of \(k\) is.

56 Parabola - Tangent Line · Level 3

(a) Show that the equation of the tangent line to the parabola \(y^2 = 4 p x\) at the point \((x_0, y_0)\) can be written as \(y_0 y = 2 p (x + x_0)\). (b) What is the \(x\)-intercept of this tangent line? Use this fact to draw the tangent line.

57 Parabola - Tangent Lines from Directrix · Level 4

Show that the tangent lines to the parabola \(x^2 = 4 p y\) drawn from any point on the directrix are perpendicular.

58 Confocal Ellipse-Hyperbola Orthogonality · Level 5

Show that if an ellipse and a hyperbola have the same foci, then their tangent lines at each point of intersection are perpendicular.

59 Numerical - Ellipse Circumference · Level 4

Use parametric equations and Simpson's Rule with \(n = 8\) to estimate the circumference of the ellipse \(9 x^2 + 4 y^2 = 36\).

60 Numerical - Pluto Orbit · Level 4

The dwarf planet Pluto travels in an elliptical orbit around the sun (at one focus). The length of the major axis is \(1.18 \times 10^{10}\) km and the length of the minor axis is \(1.14 \times 10^{10}\) km. Use Simpson's Rule with \(n = 10\) to estimate the distance traveled by the planet during one complete orbit around the sun.

61 Area - Hyperbola Segment · Level 4

Find the area of the region enclosed by the hyperbola \(x^2 / a^2 - y^2 / b^2 = 1\) and the vertical line through a focus.

62 Volume - Ellipsoid · Level 3

(a) If an ellipse is rotated about its major axis, find the volume of the resulting solid. (b) If it is rotated about its minor axis, find the resulting volume.

63 Centroid - Half Ellipse · Level 4

Find the centroid of the region enclosed by the \(x\)-axis and the top half of the ellipse \(9 x^2 + 4 y^2 = 36\).

64 Surface Area - Ellipsoid · Level 5

(a) Calculate the surface area of the ellipsoid that is generated by rotating an ellipse about its major axis. (b) What is the surface area if the ellipse is rotated about its minor axis?

65 Ellipse - Reflection Property · Level 5

Let \(P(x_1, y_1)\) be a point on the ellipse \(x^2 / a^2 + y^2 / b^2 = 1\) with foci \(F_1\) and \(F_2\) and let \(\alpha\) and \(\beta\) be the angles between the lines \(P F_1\), \(P F_2\) and the ellipse as shown in the figure. Prove that \(\alpha = \beta\). This explains how whispering galleries and lithotripsy work. Sound coming from one focus is reflected and passes through the other focus.

66 Hyperbola - Reflection Property · Level 5

Let \(P(x_1, y_1)\) be a point on the hyperbola \(x^2 / a^2 - y^2 / b^2 = 1\) with foci \(F_1\) and \(F_2\) and let \(\alpha\) and \(\beta\) be the angles between the lines \(P F_1\), \(P F_2\) and the hyperbola as shown in the figure. Prove that \(\alpha = \beta\). (This is the reflection property of the hyperbola. It shows that light aimed at a focus \(F_2\) of a hyperbolic mirror is reflected toward the other focus \(F_1\).)

67 Example - Hyperbola Standard Form · Level 2

Find the foci and asymptotes of the hyperbola \(9x^2 - 16y^2 = 144\) and sketch its graph.

68 Example - Hyperbola from Vertices and Asymptote · Level 2

Find the foci and equation of the hyperbola with vertices \((0, \pm 1)\) and asymptote \(y = 2x\).

69 Example - Shifted Ellipse · Level 2

Find an equation of the ellipse with foci \((2, -2), (4, -2)\) and vertices \((1, -2), (5, -2)\).

70 Example - Shifted Hyperbola (Completing the Square) · Level 3

Sketch the conic \(9x^2 - 4y^2 - 72x + 8y + 176 = 0\) and find its foci.

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