Basic Technical Mathematics, 11th Edition

21.9 Rotation of Axes

• Angle of Rotation of Axes • Eliminating the xy-term • Value of $B^2\hspace{0.17em}-\hspace{0.17em}4AC$ Determines Curve

In this chapter, we have discussed the circle, parabola, ellipse, and hyperbola, and in the last section, we showed how these curves are represented by the second-degree equation

$$A{x}^2\hspace{0.17em}+\hspace{0.17em}Bxy\hspace{0.17em}+\hspace{0.17em}C{y}^2\hspace{0.17em}+\hspace{0.17em}Dx\hspace{0.17em}+\hspace{0.17em}Ey\hspace{0.17em}+\hspace{0.17em}F\hspace{0.17em}=\hspace{0.17em}0$$ (21.34)

The discussion of these curves included their equations with the center (vertex of a parabola) at the origin. However, as noted in the last section, except for the special case of the hyperbola $xy\hspace{0.17em}=\hspace{0.17em}c\hspace{0.17em},\hspace{0.17em}$ we did not show what happens when the axes are rotated about the origin.

If a set of axes is rotated about the origin through an angle $\theta \hspace{0.17em},\hspace{0.17em}$ as shown in Fig. 21.97, we say that there has been a rotation of axes.