Removing Linear and Cross Terms from a Second-Degree Equation

Removing Linear and Cross Terms from a Second-Degree Equation

A general second-degree equation in two variables has the form

Two separate simplifications are usually needed to bring this to a standard form: removing the linear terms $2gx+2fy$ by translation, and removing the cross term $2hxy$ by rotation.

Removing the linear terms. Shift the origin to a point $(x_0,y_0)$ chosen so that, after substituting $x=x'+x_0,\,y=y'+y_0$, the coefficients of the new linear terms $x'$ and $y'$ both vanish. Collecting the coefficient of $x'$ and setting it to zero, and likewise for $y'$, gives the pair of simultaneous linear equations

(these are exactly the equations obtained by setting $\partial/\partial x=0$ and $\partial/\partial y=0$ on the left-hand side). Solving this pair simultaneously gives the required point $(x_0,y_0)$, and translating the origin there removes the linear terms, leaving an equation of the form $aX^2+2hXY+bY^2+c'=0$ in the new variables.

Removing the $xy$ term. Starting from an equation $aX^2+2hXY+bY^2+c'=0$, rotate the axes through an angle $\theta$ using $X=X'\cos\theta-Y'\sin\theta,\,Y=X'\sin\theta+Y'\cos\theta$ and collect the coefficient of $X'Y'$. Setting that coefficient to zero and simplifying leads to

so that

When $a=b$, the expression $\frac{2h}{a-b}$ is undefined (division by zero), which corresponds to $\tan 2\theta \to \infty$, i.e. $2\theta = 90^{\circ}$, so $\theta = 45^{\circ}$ in that special case. …