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Lecture 28

MTH 108 Lecture 28: 9.5 Rotation of Axis; General Form of Conics Notes
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Department
Mathematics
Course
MTH 108
Professor
Lun- Yi Tsai
Semester
Fall

Description
MTH 108 Precalculus Mathematics II 9.5 Rotation of Axis; General Form of Conics Notes L. Sterling Abstract Provide a generalization to each of the key terms listed in this section. Conic Equation The following would be equation for a conic: 2 2 Ax + Bxy + Cy + Dx + Ey + F = 0 A 6= 0 B 6= 0 C 6= 0 Conic Sections What are conic sections? Conic sections are various curves that would be resulting from both the intersection of both a plane and the right circular cone. Non-Rotated Conics Equation The following would be equation for a non-rotated conic, which is also known as the “general equation of a conic” or a “degenerative conic”: Ax + Cy + Dx + Ey + F = 0 A 6= 0 B = 0 C 6= 0 Graphing When it comes to graphing the general equation of a conic, then there are one of four possible graphs if the conic is non-degenerative: Conic Classification Circle A = C A 6= 0 Ellipse AC > 0 Like Signs Hyperbola AC < 0 Unlike Signs Parabola AC = 0 A 6= 0 or C 6= 0 1 Rotation of Axes Theorem What is the rotation of axes theorem? When it comes to having the x-axis and y-axis both rotate by any angle, which is labeled with ‘ ‘ ▯, to create both the x -axis and y -axis. So, when it comes to a point’s given coordinates in the xy-plane, which would be (x;y), while the a point’s given coordinates in the xy-plane, which would be (x ;y ), would create the following relationship: 2 2 Ax + Bx
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