Precalculus Mathematics II
9.5 Rotation of Axis; General Form of Conics Notes
Provide a generalization to each of the key terms listed in this section.
The following would be equation for a conic:
Ax + Bxy + Cy + Dx + Ey + F = 0
A 6= 0
B 6= 0
C 6= 0
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.
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
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:
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:
Ax + Bx