Rotate the Axes So That the New Equation Contains No $x^{2}+2 x y+y^{2}-8 x+8 y=0$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $x^{2}+2 x y+y^{2}-8 x+8 y=0$

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

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

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

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