geometric definitions and standard equations...

34
geometric definitions and standard equations of parabolas, ellipses, and hyperbolas. curves are called conic sections or conics

Upload: others

Post on 11-Jul-2020

9 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective

geometric definitions and standard equations ofparabolas, ellipses, and hyperbolas.

curves are called conic sections or conics

Page 2: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 3: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 4: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 5: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 6: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 7: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 8: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 9: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 10: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 11: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 12: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 13: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 14: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 15: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 16: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 17: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 18: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 19: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 20: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 21: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 22: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 23: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective

Reflective Propertiesapplications of parabolas involve their use asreflectors of light and radio waves.Rays originating at a parabola’s focus are reflected out ofthe parabola parallel to the parabola’s axis

Page 24: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective

If an ellipse is revolved about itsmajor axis to generate a surface (thesurface is called an ellipsoid ) andthe interior is silvered to produce amirror, light from one focus will bereflected to the other focusEllipsoids reflect sound the sameway, and this property is used toconstruct whispering galleries,rooms in which a person standing atone focus can hear a whisper fromthe other focus.

Page 25: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 26: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 27: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 28: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 29: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 30: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 31: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 32: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 33: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective
Page 34: geometric definitions and standard equations ofmimoza.marmara.edu.tr/~cem/Calculus2/C2_3.pdfparabolas, ellipses, and hyperbolas. curves are called conicsectionsor conics Reflective

Replace the Cartesian equations by equivalent polar equations