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

Multiple Choice

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

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

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

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