Basic Technical Mathematics, 11th Edition

21.7 Translation of Axes

• Centers of Curves Not at Origin • Axes of Curves Parallel to Coordinate Axes • Translation of Axes

The equations we have considered for the parabola, the ellipse, and the hyperbola are those for which the center of the ellipse or hyperbola, or vertex of the parabola, is at the origin. In this section, we consider, without specific use of the definition, the equations of these curves for the cases in which the axis of the curve is parallel to one of the coordinate axes. This is done by translation of axes.

In Fig. 21.83, we choose a point (h, k) in the xy-coordinate plane as the origin of another coordinate system, the $x^{\hspace{0.17em}\prime \hspace{0.17em}}{y}^{\hspace{0.17em}\prime \hspace{0.17em}}\text{ - coordinate}$ system. The $x^{\hspace{0.17em}\prime \hspace{0.17em}}\text{ - axis}$ is parallel to the x-axis and the $y^{\hspace{0.17em}\prime \hspace{0.17em}}\text{ - axis}$ is parallel ...