91Ó°ÊÓ

An equation is$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,B≠0$.

The rotation of axes through an appropriate angle $\mathrm{θ}$will transform the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$into an equation in $x\text{'}$and $y^{\text{'}}$without an $x^{2}y$term.

In order to find the formula for choosing an appropriate angle $\mathrm{θ}$, we use the rotation formulas in the above equation.

$A{\left(x\text{'}\cos \mathrm{θ}-y\text{'}\sin \mathrm{θ}\right)}^2+B\left(x\text{'}\cos \mathrm{θ}-y\text{'}\sin \mathrm{θ}\right)\left(x\text{'}\sin \mathrm{θ}+y\text{'}\cos \mathrm{θ}\right)+C{\left(x\text{'}\sin \mathrm{θ}+y\text{'}\cos \mathrm{θ}\right)}^2+D\left(x\text{'}\cos \mathrm{θ}-y\text{'}\sin \mathrm{θ}\right)+E\left(x\text{'}\sin \mathrm{θ}+y\text{'}\cos \mathrm{θ}\right)+F=0$

Expand and collecting the like terms.$\left(A{\cos }^2\mathrm{θ}+B\sin \mathrm{θ}\cos \mathrm{θ}+C{\sin }^2\mathrm{θ}\right){\left(x\text{'}\right)}^2+\left[B\left({\cos }^2\mathrm{θ}-{\sin }^2\mathrm{θ}+2\left(C-A\right)\sin \mathrm{θ}\cos \mathrm{θ}\right)\right]x\text{'}y\text{'}+\left(A{\sin }^2\mathrm{θ}-B\sin \mathrm{θ}\cos \mathrm{θ}+C{\cos }^2\mathrm{θ}\right){\left(y\text{'}\right)}^2+\left(D\cos \mathrm{θ}+E\sin \mathrm{θ}\right)x\text{'}+\left(-D\sin \mathrm{θ}+E\cos \mathrm{θ}\right)y\text{'}+F=0$Since we want to eliminate the $x^{2}y$term from this equation, we select an angle $\mathrm{θ}$, so that the coefficient of the $x^{2}y$term, $B\left({\cos }^2\mathrm{θ}-{\sin }^2\mathrm{θ}\right)+2(C-A)\sin \mathrm{θ}\cos \mathrm{θ}$is $0$.

$\begin{array}{r}B\left({\cos }^2\mathrm{θ}-{\sin }^2\mathrm{θ}\right)+2(C-A)\sin \mathrm{θ}\cos \mathrm{θ}=0\\ B\cos \left(2\mathrm{θ}\right)+(C-A)\sin \left(2\mathrm{θ}\right)=0\end{array}$

role="math" localid="1646661431364" $\begin{array}{rcl}B\cos \left(2\mathrm{θ}\right)& =& (A-C)\sin \left(2\mathrm{θ}\right)\\ \cot \left(2\mathrm{θ}\right)& =& \frac{A-C}{B};B≠0\end{array}$

Therefore, to transform the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,B≠0$into one and$y\text{'}$ without an$x^{2}y$term, rotate the axes through an acute angle $\mathrm{θ}$that satisfies the equation $\cot \left(2\mathrm{θ}\right)=\frac{A-C}{B}$