Rotate the Axes So That the New Equation Contains No $$x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $$x ^ { 2 } + x y + y ^ { 2 } - 3 y - 6 = 0$$

A)
$\begin{array}{l}\theta=45^{\circ} \\y^{\prime 2}=-18 x^{\prime} \\\text { parabola } \\\text { vertex at }(0,0) \\\text { focus at }\left(-\frac{9}{2}, 0\right) \end{array}$

B)
$$\frac{\theta=45^{\circ}}{\left(x^{\prime}-\frac{\sqrt{2}}{2}\right) ^{2}}+\frac{\left(y^{\prime}-\frac{3 \sqrt{2}}{2}\right) ^{2}}{15}=1$$
ellipse center at $\left(\frac{\sqrt{2}}{2}, \frac{3 \sqrt{2}}{2}\right)$
major axis is $y^{\prime}$ -axis vertices at $\left(\frac{\sqrt{2}}{2},-\frac{3 \sqrt{2}}{2}\right)$ and $\left(\frac{\sqrt{2}}{2}, \frac{9 \sqrt{2}}{2}\right)$

C) $$\theta=45^{\circ}$$
$$\frac{x^{\prime 2}}{6}-\frac{y^{\prime 2}}{8}=1$$
hyperbola
center at $(0,0)$
transverse axis is the $\mathrm{x}^{\prime}$ -axis
vertices at $(\pm \sqrt{6}, 0)$

D) $$\begin{array}{l}\theta=45^{\circ} \\\frac{x^{\prime 2}}{3}+\frac{y^{2}}{4}=1\end{array}$$
ellipse
center at $(0,0)$
major axis is $y^{\prime}$ -axis
vertices at $(0, \pm 2)$