Summary

Here is an overall summary of the use of rotations and translations of axes. The problem is to graph an equation of the form

Ax² + Bxy + Cy² + Dx + Ey + F = 0.

By Rotation of Axes, we get a new equation of the simpler form

A₁X² + C₁Y² + D₁X + E₁Y + F₁ = 0.

If either A₁ = 0 or C₁ = 0, the curve is a parabola that can be sketched by the method of Section 5.4. If A₁ and C₁ are both nonzero, Translation of Axes gives us a new equation of the simpler form

A₂ U² + B₂ V² + F₂ = 0.

The graph of this equation is an ellipse or hyperbola, which can be sketched by the method of Section 5.5. The degenerate cases — two lines, one line, a point, or an empty graph — may also occur.