Chapter 6 – Analyzing Conic Sections Answer Key6.1 Equation of an Ellipse

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6.2 Focal Property of Ellipses

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This is a circle with a radius of 2, centered at (-4, -2)This is a circle with a radius of 4, centered at (-2, 0)This is a circle with a radius of 1, centered at (3, 2)This is a circle with a radius of 3, centered at (-2, 1)

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11.12.13.14.

This is a circle with a radius of 3, centered at (-2, 1)This is a circle with a radius of 3, centered at (3, 1)While they are technically ellipses, these last 5 are circles, in the same way that squaresare technically special cases of rectangles

billion miles.

The area of an ellipse is . To see why this is true, start with a circle of radius 1,which has an area of . Then imagine an approximation with rectangles of the circle.Then stretch the rectangles by a factor of in the direction and by a factor of in the

direction to obtain an approximation of the ellipse. This makes the rectangles timeswider and times taller, giving an area that is multiplied by the area of theapproximation of the circle. Since this is true of any approximation of the circle, the areaof the ellipse must be

The echo room has a major axis of 100 m and a minor axis of 34.12 m.

Situating the room in the coordinate plane, the room can be represented by the equation:

. You will be 94 m from the person you are spying on.

Answers will vary.

6.3 Parabolas and the Distance Formula

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Distance = 30.53Distance = 10.63Distance = 8.06Distance = 6.32Distance = 23.19

Coordinates: (-.4, 0) | (1, 0)

Distance =

Coordinates: (1/4, 0) | (2/3, 0)

Distance =

Coordinates: (2, 0) | (4, 0)Distance = 2

Coordinates: (-.75, 0) | (-1.5, 0)Distance = .75

Coordinates: (-1 1/3, 0) | (-.75, 0) Distance =

6.4 Parabolas and Analytic Geometry

AnswersadbecFocus (0, p)Directrix:Focal radiusAxis of symmetry:Vertex: (0, 0)

Vertex: (-2, 1)

Axis of Symmetry: Equation:

Vertex: (2, 4)

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1.2.3.4.5.6.7.8.9.10.11.12.13.

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Vertex: (2, 4)Axis of Symmetry: Equation:

Vertex: (-4, 4)Axis of Symmetry: Equation:

6.5 Applications of Parabolas

AnswersIf a ball is thrown from an initial height of 200 feet with an initial velocity of 96 feet persecond, after 3 seconds it will reach a maximum height of 344 feet.

B. 628 feet (apx 6 sec@ avg 100ft/sec)C. 36.64 feeta) 2304 ft

b) 256 ftc) 6 seconds and 10 secondsd) 21 seconds

16 feet (the vertex of the parabola)The focus is (0, 91/4), the directrix is y=109/4 : The depth of the dish is 8.9 metersA) b) (1/2 way)

gives the height in cm, multiply by 2 to get the height in meters.

19 cables5, 5.45, 6.8, 9.05, 12.2, 16.25, 21.2, 27.05,

33.8, 41.45$2199.00 – Specialty Cable $2081.60 – Cables R Us

yes, answers will varyAnswers will vary, should note that it appears that rays parallel to the x-axis will alwaysbe reflected to strike the focus, where the receiver is mounted on a parabolic dish.

6.6 Hyperbola Equations and the Focal Property

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6.7 Hyperbolas and Asymptotes

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6.8 Conic Sections and Dandelin Spheres

AnswersGerminal Dandelin : 1822

The focal property and location of foci

The loci of the point of contact between the spheres and the cutting plane (the imagebelow illustrates this)

The line segments are equal in length.

Tangents are perpendicular to radii of the sphere

A line drawn from one focus to any point on the ellipse and back to the other focus is thesame length as a similar line drawn to any other point on the ellipse.

Trick question, he did not. Pierce Morton did, using Dandelin’s Spheres, 1829.

The directrix line

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The directrix line

The focus

The ellipse

The parabola

The hyperbola

– 21. Refer to image below:

The conic section created is an ellipse.

6.9 General Forms of Conic Sections

AnswersIdentify the conic section1. Ellipse 2. circle 3. hyperbola 4. parabola Convert to standard form and identify the conic section

5. it is an ellipse

6. it is a hyperbola.7. it is a circle.

8. it is an ellipse

9. it is an ellipse

10. it is a hyperbola

Graph:11. : 12. :

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CK-12 Math Analysis Concepts 1