Conic Sections/Rotation of Axes

Introduction

A conic is a second degree polynomial. When it is expanded, you get an equation of the form: $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$ . If the value of $B$ is zero then the conic is not rotated and sits on the x- and y- axes. If $B$ is non-zero, then the conic is rotated about the axes, with the rotation centred on the origin.

Graphing

Graphing a Rotated Conic

If you are asked to graph a rotated conic in the form $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$ , it is first necessary to transform it to an equation for an identical, non-rotated conic. This is then plotted onto new axes which are drawn onto the graph. The equation for the nonrotated conic can be found by: $AX^{2}+BXY+CY^{2}+DX+EY+F=0$ . Note how capital letters are used for the pronumerals. This signifies that they represent different values to the original equation.

To determine the values of X and Y, you use the formulae:

$\tan 2\theta ={\frac {B}{A-C}}$
$x=X\cos \theta -Y\sin \theta$
$y=X\sin \theta +Y\cos \theta$

By substituting the new values of $x$ and $y$ into the original equation, a new one can be obtained which represents a non-rotated conic which can be plotted on a set of axes rotated at $\theta$ (anti-clockwise) to the original x- and y- axes. When you do this, however, it will still be necessary to determine the new rotated location of points such as the vertex, foci and directrixes. This can be done using the following formulae:

$X=x\cos \theta +y\sin \theta$
$Y=-x\sin \theta +y\cos \theta$

Where (X,Y) are the new rotated coordinates of the original point (x,y).

The formula: $B^{2}-4AC$ can be used to determine the type of conic from the original equation before you start graphing:

$B^{2}-4AC=0$ : Parabola
$B^{2}-4AC<0$ : Ellipse
$B^{2}-4AC>0$ : Hyperbola

Rotating a Conic

If you wish to rotate a conic by a certain angle, $\theta$ , it is relatively simple. All you do is make the following substitution from the previous section:

$x=X\cos \theta -Y\sin \theta$
$y=X\sin \theta +Y\cos \theta$

Replacing the $x$ and $y$ values from the function with these new ones. Then simplify the answer, and it will be a function for the same conic rotated by $\theta$ counter-clockwise about the origin.

References