Rotate the Axes So That the New Equation Contains No $x y + 16 = 0$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $x y + 16 = 0$

A) $$\theta=45^{\circ}$$
$$\frac{y^{\prime 2}}{32}+\frac{x^{\prime 2}}{32}=1$$
ellipse
center at $(0,0)$
major axis is $y^{\prime}$ -axis
vertices at $(0, \pm 4 \sqrt{2})$

B) $$\theta=45^{\circ}$$
$$\frac{y^{\prime 2}}{32}-\frac{x^{\prime 2}}{32}=1$$
hyperbola
center at $(0,0)$
transverse axis is $\mathrm{y}^{\prime}$ -axis
vertices at $(0, \pm 4 \sqrt{2})$

C) $\begin{array}{l}\theta=45^{\circ} \\y^{\prime 2}=-32 x^{\prime} \\\text { parabola } \\\text { vertex at }(0,0) \\\text { focus at }(-8,0) \end{array}$

D) $$\begin{array}{l}\theta=36.9^{0} \\\frac{x^{12}}{4}+\frac{y^{\prime 2}}{2}=1\end{array}$$
ellipse
center at $(0,0)$
major axis is the $x^{\prime}$ -axis
vertices at $(\pm 2,0)$