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Conic Sections Hyperbolas

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Definition The conic section formed by a plane which intersects both of the right conical surfaces Formed when or when the plane is parallel to the axis of the cone

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Definition A hyperbola is the set of all points in the plane where The difference between the distances From two fixed points (foci) Is a constant Geogebra Demonstration Geogebra Demonstration

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Experimenting with Definition Turn on Explore geometric definition. A purple point will appear on the hyperbola, along with two line segments labeled L1 and L2. Drag the purple point around the hyperbola.Explore geometric definition Do the lengths of L1 and L2 change? What do you notice about the absolute value of the differences of the lengths? How do these observations relate to the geometric definition of a hyperbola. Observe the values of L1, L2, and the difference | L1 − L2 | as you vary the values of a and b. How is the difference | L1 − L2 | related to the values of a and/or b? (Hint: Think about multiples.) Determine the difference | L1 − L2 | for a hyperbola where a = 3 and b = 4. Use the Gizmo to check your answer.

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Elements of An Ellipse Transverse axis Line joining the intercepts Conjugate axis Passes through center, perpendicular to transverse axis Vertices Points where hyperbola intersects transverse axis

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Elements of An Ellipse Transverse Axis Length = 2a Foci Location (-c, 0), (c, 0) Asymptotes Experiment with Pythagorean relationship Experiment with Pythagorean relationship

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Equations of An Ellipse Given equations of ellipse Centered at origin Opening right and left Equations of asymptotes Opening up and down Equations of asymptotes

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Try It Out Find The center Vertices Foci Asymptotes

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More Trials and Tribulations Find the equation in standard form of the hyperbola that satisfies the stated conditions. Vertices at (0, 2) and (0, -2), foci (0, 3) and (0, -3) Foci (1, -2) and (7, -2) slope of an asymptote = 5/4

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Assignment Hyperbola A Exercise set 6.3 Exercises 1 – 25 odd and 33 – 45 odd

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Conic Sections

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Eccentricity of a Hyperbola The hyperbola can be wide or narrow

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Eccentricity of a Hyperbola As with eccentricity of an hyperbola, the formula is Note that for hyperbolas c > a Thus eccentricity > 1

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Try It Out If the vertices are (1, 6) and (1, 8) and the eccentricity is 5/2 Find the equation (standard form) of the hyperbola The center of the hyperbola is at (-3, -3) and the conjugate axis has length 6, and the eccentricity = 2 Find two possible hyperbola equations

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Application – Locating Position For any point on a hyperbolic curve Difference between distances to foci is constant. Result: hyperbolas can be used to locate enemy guns “If the sound of an enemy gun is heard at two listening posts and the difference in time is calculated, then the gun is known to be located on a particular hyperbola. A third listening post will determine a second hyperbola, and then the gun emplacement can be spotted as the intersection of the two hyperbolas.”

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Application – Locating Position The loran system navigator equipped with a map that gives curves, called loran lines of position. Navigators find the time interval between these curves, Narrow down the area that their craft's position is in. Then switch to a different pair of loran transmitters Repeat the procedure Find another curve representing the craft's position.

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Construction Consider a the blue string Keep marker against ruler and with string tight Keep end of ruler on focus F 1, string tied to other end

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Graphing a Hyperbola on the TI As with the ellipse, the hyperbola is not a function Possible to solve for y Get two expressions Graph each What happens if it opens right and left?

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Graphing a Hyperbola on the TI Top and bottom of hyperbola branches are graphed separately As with ellipses you must Ellipse with Geogebra

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Assignment Hyperbolas 2 Exercise Set 6.3 Exercises 27 – 31 odd and 49 – 63 odd

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