Rotate the Axes So That the New Equation Contains No $31 x^{2}+10 \sqrt{3} x y+21 y^{2}-144=0$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $31 x^{2}+10 \sqrt{3} x y+21 y^{2}-144=0$

A)
$$\begin{array}{l}\theta=45^{\circ} \\x^{\prime 2}=-4 \sqrt{2} y^{\prime}\end{array}$$
parabola
vertex at $(0,0)$
focus at $(0,-\sqrt{2})$

B)
$\begin{array}{l}\theta=36.9^{\circ} \\\frac{x^{2}}{9}+\frac{y^{\prime 2}}{4}=1 \\\text { ellipse } \\\text { center at }(0,0) \\\text { major axis is } x^{\prime} \text {-axis } \\\text { vertices at }(\pm 3.0) \end{array}$

C)
$$\begin{array}{l}\theta=30^{0} \\\frac{x^{2}}{4}+\frac{y^{+2}}{9}=1\end{array}$$
ellipse
center at $(0,0)$
major axis is $y^{\prime}$ -axi:
vertices at $(0, \pm 3)$

D)
$\begin{array}{l}\theta=45^{\circ} \\y^{\prime 2}=-4 \sqrt{2} x^{\prime} \\\text { parabola } \\\text { vertex at }(0,0) \\\text { focus at }(-\sqrt{2}, 0) \end{array}$