10.4 Rotation of Conics

In this section, we learned how to find the equation and sketch the graph of a conic section that is not perfectly vertical or horizontal, but on tilted axes. The following picture is of three equations that are necessary to start every one of these problems:

First, you must use the top equation to find the angle of rotation of the axes. You then use that angle to simplify the bottom two equations. The ultimate goal is to replace x and y in the original conic's equation with x' and y', or the new x and y that correspond with the new axes. The next picture shows the classifications of each type of conic section using its determinant. From the value of the determinant, we can tell which type of conic section an equation represents before completing the problem. This will be helpful to double-check the result.