Translation of Axes

We now turn to the method of Translation of Axes. This method is useful for graphing a second degree equation with no xy-term,

Ax² + Cy² + Dx + Ey + F = 0.

If A or C is zero, the graph will be a horizontal or vertical parabola, which can be graphed by the method of Section 5.4. If both A and C are nonzero, the graph turns out to be an ellipse or hyperbola with horizontal and vertical axes X and Y, as in Figure 5.6.1. In the method of Translation of Axes, we take X and Y as a new pair of coordinate axes and get a new equation for the curve in the simple form

AX² + CY² + F₁ = 0.

Figure 5.6.1

This curve can be sketched as in Section 5.5. The name "Translation of Axes" means that the original coordinate axes x and y are replaced by new coordinate axes X and y, which are parallel to the original axes.

The new axes are found using a procedure from algebra called "completing the squares." This procedure changes an expression like Ax² + Dx into a perfect square plus a constant.