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4. Standard Equations of Hyperbolas Applications and Problem-Solving The Hyperbola Conic Sections > The Hyperbola Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra?campaign_content=book_196_section_51&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. Similar to a parabola, a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an ellipse does. A hyperbola consists of two disconnected curves called its arms or branches. If the transverse axis of any hyperbola is aligned with the x-axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as x2/a2 - y2/b2 = 1. Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola" and has equation y2/a2 - x2/b2 = 1. Standard Equations of Hyperbolas Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/conic-sections-7/the-hyperbola-51/standard-equations-of-hyperbolas-219- 5872?campaign_content=book_196_section_51&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Properties of a Hyperbola View on Boundless.com Conic Sections > The Hyperbola

6. Hyperbolas have applications to a number of different systems and problems including sundials and trilateration. Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. A hyperbola is the basis for solving trilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Applications and Problem-Solving Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/conic-sections-7/the-hyperbola-51/applications-and-problem-solving-220- 11100?campaign_content=book_196_section_51&campaign_term=Algebra&utm_campaign=powerpoint&utm_medium=direct&utm_source=boun dless Hyperbolas and Sundials View on Boundless.com Conic Sections > The Hyperbola

7. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

8. Key terms asymptote A line that a curve approaches arbitrarily closely, as they go to infinity; the limit of the curve, its tangent "at infinity". conic section Any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola. hyperbola A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone. parabola The conic section formed by the intersection of a cone with a plane parallel to a plane tangent to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix). trilateration The determination of the location of a point based on its distance from three other points. Free to share, print, make copies and changes. Get yours at www.boundless.com Conic Sections

9. Properties of a Hyperbola The asymptotes of the hyperbola (red curves) are shown as blue dashed lines and intersect at the center of the hyperbola, C. The two focal points are labeled F1 and F2, and the thin black line joining them is the transverse axis. The perpendicular thin black line through the center is the conjugate axis. The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2. The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Hyperbola." GNU FDL http://en.wikipedia.org/wiki/Hyperbola View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/HyperbolaView on Boundless.com Conic Sections

10. Hyperbola A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane may or may not be parallel to the axis of the cone Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Hyperbola (PSF)." Public domain http://en.wikipedia.org/wiki/File:Hyperbola_(PSF).png View on Boundless.comPublic domainhttp://en.wikipedia.org/wiki/File:Hyperbola_(PSF).pngView on Boundless.com Conic Sections

11. Kepler Orbits A diagram of the various forms of the Kepler Orbit and their eccentricities. Blue is a hyperbolic trajectory (e > 1). Green is a parabolic trajectory (e = 1). Red is an elliptical orbit (e < 1). Grey is a circular orbit (e = 0). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "OrbitalEccentricityDemo." CC BY-SA http://en.wikipedia.org/wiki/File:OrbitalEccentricityDemo.svg View on Boundless.comCC BY-SAhttp://en.wikipedia.org/wiki/File:OrbitalEccentricityDemo.svgView on Boundless.com Conic Sections

12. Hyperbolas and Sundials Hyperbolas as declination lines on a sundial. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Hyperbola." GNU FDL http://en.wikipedia.org/wiki/Hyperbola View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/HyperbolaView on Boundless.com Conic Sections

13. Free to share, print, make copies and changes. Get yours at www.boundless.com Conic Sections An example of trilateration is: A) A ship locating its position using arrival times of signals from GPS transmitters. B) The path a particle takes if its total energy is greater than zero. C) The Earth's orbit around the sun. D) The shadow created by the sun and a sundial.

14. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ Conic Sections An example of trilateration is: A) A ship locating its position using arrival times of signals from GPS transmitters. B) The path a particle takes if its total energy is greater than zero. C) The Earth's orbit around the sun. D) The shadow created by the sun and a sundial.

15. Attribution Wikipedia. "Hyperbola." CC BY-SA 3.0 http://en.wikipedia.org/wiki/HyperbolaCC BY-SA 3.0http://en.wikipedia.org/wiki/Hyperbola Wiktionary. "asymptote." CC BY-SA 3.0 http://en.wiktionary.org/wiki/asymptoteCC BY-SA 3.0http://en.wiktionary.org/wiki/asymptote Wiktionary. "hyperbola." CC BY-SA 3.0 http://en.wiktionary.org/wiki/hyperbolaCC BY-SA 3.0http://en.wiktionary.org/wiki/hyperbola Wiktionary. "parabola." CC BY-SA 3.0 http://en.wiktionary.org/wiki/parabolaCC BY-SA 3.0http://en.wiktionary.org/wiki/parabola Wikipedia. "Hyperbola." CC BY-SA 3.0 http://en.wikipedia.org/wiki/HyperbolaCC BY-SA 3.0http://en.wikipedia.org/wiki/Hyperbola Wikipedia. "Kepler orbit." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Kepler_orbitCC BY-SA 3.0http://en.wikipedia.org/wiki/Kepler_orbit Wiktionary. "conic section." CC BY-SA 3.0 http://en.wiktionary.org/wiki/conic+sectionCC BY-SA 3.0http://en.wiktionary.org/wiki/conic+section Wiktionary. "trilateration." CC BY-SA 3.0 http://en.wiktionary.org/wiki/trilaterationCC BY-SA 3.0http://en.wiktionary.org/wiki/trilateration Wiktionary. "hyperbola." CC BY-SA 3.0 http://en.wiktionary.org/wiki/hyperbolaCC BY-SA 3.0http://en.wiktionary.org/wiki/hyperbola Free to share, print, make copies and changes. Get yours at www.boundless.com Conic Sections