Stewart Precalc 6e Section 11.2: Ellipses

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Stewart Precalc 6e Section 11.2: Ellipses 0/63
1 Concepts - Definition · Level 1
An ellipse is the set of all points in the plane for which the _____ of the distances from two fixed points \(F_1\) and \(F_2\) is constant. The points \(F_1\) and \(F_2\) are called the _____ of the ellipse.
2 Concepts - Standard form (horizontal) · Level 1
The graph of the equation \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) with \(a > b > 0\) is an ellipse with vertices (__, __) and (__, __) and foci \((\pm c, 0)\), where \(c = \) _____. So the graph of \(\dfrac{x^2}{5^2} + \dfrac{y^2}{4^2} = 1\) is an ellipse with vertices (__, __) and (__, __) and foci (__, __) and (__, __).
3 Concepts - Standard form (vertical) · Level 1
The graph of the equation \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\) with \(a > b > 0\) is an ellipse with vertices (__, __) and (__, __) and foci \((0, \pm c)\), where \(c = \) _____. So the graph of \(\dfrac{x^2}{4^2} + \dfrac{y^2}{5^2} = 1\) is an ellipse with vertices (__, __) and (__, __) and foci (__, __) and (__, __).
4 Concepts - Label graph · Level 1
Label the vertices and foci on the graphs given for the ellipses in Exercises 2 and 3. (a) \(\dfrac{x^2}{5^2} + \dfrac{y^2}{4^2} = 1\) (b) \(\dfrac{x^2}{4^2} + \dfrac{y^2}{5^2} = 1\)
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5 Skills - Match equation with graph · Level 2
Match the equation \(\dfrac{x^2}{16} + \dfrac{y^2}{4} = 1\) with one of the graphs labeled I-IV. Give reasons for your answer.
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6 Skills - Match equation with graph · Level 2
Match the equation \(x^2 + \dfrac{y^2}{9} = 1\) with one of the graphs labeled I-IV. Give reasons for your answer.
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7 Skills - Match equation with graph · Level 2
Match the equation \(4x^2 + y^2 = 4\) with one of the graphs labeled I-IV. Give reasons for your answer.
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8 Skills - Match equation with graph · Level 2
Match the equation \(16x^2 + 25y^2 = 400\) with one of the graphs labeled I-IV. Give reasons for your answer.
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9 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1\). Determine the lengths of the major and minor axes, and sketch the graph.
10 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1\). Determine the lengths of the major and minor axes, and sketch the graph.
11 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(9x^2 + 4y^2 = 36\). Determine the lengths of the major and minor axes, and sketch the graph.
12 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(4x^2 + 25y^2 = 100\). Determine the lengths of the major and minor axes, and sketch the graph.
13 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(x^2 + 4y^2 = 16\). Determine the lengths of the major and minor axes, and sketch the graph.
14 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(4x^2 + y^2 = 16\). Determine the lengths of the major and minor axes, and sketch the graph.
15 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(2x^2 + y^2 = 3\). Determine the lengths of the major and minor axes, and sketch the graph.
16 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(5x^2 + 6y^2 = 30\). Determine the lengths of the major and minor axes, and sketch the graph.
17 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(x^2 + 4y^2 = 1\). Determine the lengths of the major and minor axes, and sketch the graph.
18 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(9x^2 + 4y^2 = 1\). Determine the lengths of the major and minor axes, and sketch the graph.
19 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(\dfrac{1}{2} x^2 + \dfrac{1}{8} y^2 = \dfrac{1}{4}\). Determine the lengths of the major and minor axes, and sketch the graph.
20 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(x^2 = 4 - 2y^2\). Determine the lengths of the major and minor axes, and sketch the graph.
21 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(y^2 = 1 - 2x^2\). Determine the lengths of the major and minor axes, and sketch the graph.
22 Skills - Find vertices, foci, eccentricity · Level 2
Find the vertices, foci, and eccentricity of the ellipse \(20x^2 + 4y^2 = 5\). Determine the lengths of the major and minor axes, and sketch the graph.
23 Skills - Find equation from graph · Level 2
Find an equation for the ellipse whose graph is shown.
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24 Skills - Find equation from graph · Level 2
Find an equation for the ellipse whose graph is shown.
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25 Skills - Find equation from graph · Level 2
Find an equation for the ellipse whose graph is shown.
26 Skills - Find equation from graph · Level 2
Find an equation for the ellipse whose graph is shown.
27 Skills - Find equation from graph · Level 2
Find an equation for the ellipse whose graph is shown.
28 Skills - Find equation from graph · Level 2
Find an equation for the ellipse whose graph is shown.
29 Skills - Graph with technology · Level 2
Use a graphing device to graph the ellipse \(\dfrac{x^2}{25} + \dfrac{y^2}{20} = 1\).
30 Skills - Graph with technology · Level 2
Use a graphing device to graph the ellipse \(x^2 + \dfrac{y^2}{12} = 1\).
31 Skills - Graph with technology · Level 2
Use a graphing device to graph the ellipse \(6x^2 + y^2 = 36\).
32 Skills - Graph with technology · Level 2
Use a graphing device to graph the ellipse \(x^2 + 2y^2 = 8\).
33 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Foci: \((\pm 4, 0)\), vertices: \((\pm 5, 0)\).
34 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Foci: \((0, \pm 3)\), vertices: \((0, \pm 5)\).
35 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Length of major axis: 4, length of minor axis: 2, foci on \(y\)-axis.
36 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Length of major axis: 6, length of minor axis: 4, foci on \(x\)-axis.
37 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Foci: \((0, \pm 2)\), length of minor axis: 6.
38 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Foci: \((\pm 5, 0)\), length of major axis: 12.
39 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Endpoints of major axis: \((\pm 10, 0)\), distance between foci: 6.
40 Skills - Find equation from conditions · Level 2
Find an equation for the ellipse that satisfies the given conditions. Endpoints of minor axis: \((0, \pm 3)\), distance between foci: 8.
41 Skills - Find equation from conditions · Level 3
Find an equation for the ellipse that satisfies the given conditions. Length of major axis: 10, foci on \(x\)-axis, ellipse passes through the point \((\sqrt{5}, 2)\).
42 Skills - Find equation from conditions · Level 3
Find an equation for the ellipse that satisfies the given conditions. Eccentricity: \(\dfrac{1}{9}\), foci: \((0, \pm 2)\).
43 Skills - Find equation from conditions · Level 3
Find an equation for the ellipse that satisfies the given conditions. Eccentricity: 0.8, foci: \((\pm 1.5, 0)\).
44 Skills - Find equation from conditions · Level 3
Find an equation for the ellipse that satisfies the given conditions. Eccentricity: \(\dfrac{\sqrt{3}}{2}\), foci on \(y\)-axis, length of major axis: 4.
45 Skills - Intersection of ellipses · Level 3
Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. \(\begin{cases} 4x^2 + y^2 = 4 \\ 4x^2 + 9y^2 = 36 \end{cases}\)
46 Skills - Intersection of ellipses · Level 3
Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. \(\begin{cases} \dfrac{x^2}{16} + \dfrac{y^2}{9} = 1 \\ \dfrac{x^2}{9} + \dfrac{y^2}{16} = 1 \end{cases}\)
47 Skills - Intersection of ellipses · Level 3
Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. \(\begin{cases} 100 x^2 + 25 y^2 = 100 \\ x^2 + \dfrac{y^2}{9} = 1 \end{cases}\)
48 Skills - Ancillary circle · Level 3
The ancillary circle of an ellipse is the circle with radius equal to half the length of the minor axis and center the same as the ellipse (see the figure). The ancillary circle is thus the largest circle that can fit within an ellipse. (a) Find an equation for the ancillary circle of the ellipse \(x^2 + 4y^2 = 16\). (b) For the ellipse and ancillary circle of part (a), show that if \((s, t)\) is a point on the ancillary circle, then \((2s, t)\) is a point on the ellipse.
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49 Skills - Family of ellipses · Level 3
(a) Use a graphing device to sketch the top half (the portion in the first and second quadrants) of the family of ellipses \(x^2 + k y^2 = 100\) for \(k = 4, 10, 25\), and 50. (b) What do the members of this family of ellipses have in common? How do they differ?
50 Skills - Family with same foci · Level 3
If \(k > 0\), the following equation represents an ellipse: \(\dfrac{x^2}{k} + \dfrac{y^2}{4 + k} = 1\). Show that all the ellipses represented by this equation have the same foci, no matter what the value of \(k\).
51 Applications - Earth orbit · Level 3
Perihelion and Aphelion. The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. The earth's distance from the sun is 147,000,000 km at perihelion and 153,000,000 km at aphelion. Find an equation for the earth's orbit. (Place the origin at the center of the orbit with the sun on the \(x\)-axis.)
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52 Applications - Pluto orbit · Level 3
The Orbit of Pluto. With an eccentricity of 0.25, Pluto's orbit is the most eccentric in the solar system. The length of the minor axis of its orbit is approximately 10,000,000,000 km. Find the distance between Pluto and the sun at perihelion and at aphelion. (See Exercise 51.)
53 Applications - Lunar orbit · Level 3
Lunar Orbit. For an object in an elliptical orbit around the moon, the points in the orbit that are closest to and farthest from the center of the moon are called perilune and apolune, respectively. These are the vertices of the orbit. The center of the moon is at one focus of the orbit. The Apollo 11 spacecraft was placed in a lunar orbit with perilune at 68 mi and apolune at 195 mi above the surface of the moon. Assuming that the moon is a sphere of radius 1075 mi, find an equation for the orbit of Apollo 11. (Place the coordinate axes so that the origin is at the center of the orbit and the foci are located on the \(x\)-axis.)
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54 Applications - Plywood ellipse · Level 2
Plywood Ellipse. A carpenter wishes to construct an elliptical table top from a sheet of plywood, 4 ft by 8 ft. He will trace out the ellipse using the thumbtack and string method illustrated in Figures 2 and 3. What length of string should he use, and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut out of the plywood sheet?
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55 Applications - Sunburst window · Level 2
Sunburst Window. A sunburst window above a doorway is constructed in the shape of the top half of an ellipse, as shown in the figure. The window is 20 in. tall at its highest point and 80 in. wide at the bottom. Find the height of the window 25 in. from the center of the base.
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56 Discovery - Drawing ellipse · Level 2
Drawing an Ellipse on a Blackboard. Try drawing an ellipse as accurately as possible on a blackboard. How would a piece of string and two friends help this process?
57 Discovery - Light cone · Level 2
Light Cone from a Flashlight. A flashlight shines on a wall, as shown in the figure. What is the shape of the boundary of the lighted area? Explain your answer.
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58 Discovery - Latus rectum · Level 3
How Wide Is an Ellipse at Its Foci? A latus rectum for an ellipse is a line segment perpendicular to the major axis at a focus, with endpoints on the ellipse, as shown in the figure. Show that the length of a latus rectum is \(2 b^2 / a\) for the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) with \(a > b\).
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59 Discovery - Paper around bottle · Level 2
Is It an Ellipse? A piece of paper is wrapped around a cylindrical bottle, and then a compass is used to draw a circle on the paper, as shown in the figure. When the paper is laid flat, is the shape drawn on the paper an ellipse? (You don't need to prove your answer, but you might want to do the experiment and see what you get.)
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60 Example - Sketching an Ellipse · Level 2
An ellipse has the equation \(\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1\). (a) Find the foci, the vertices, and the lengths of the major and minor axes, and sketch the graph. (b) Draw the graph using a graphing calculator.
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61 Example - Finding the Foci of an Ellipse · Level 2
Find the foci of the ellipse \(16 x^2 + 9 y^2 = 144\), and sketch its graph.
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62 Example - Finding the Equation of an Ellipse · Level 2
The vertices of an ellipse are \((\pm 4, 0)\), and the foci are \((\pm 2, 0)\). Find its equation, and sketch the graph.
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63 Example - Finding the Equation of an Ellipse from Its Eccentricity and Foci · Level 3
Find the equation of the ellipse with foci \((0, \pm 8)\) and eccentricity \(e = \dfrac{4}{5}\), and sketch its graph.
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