Rotate the Axes So That the New Equation Contains No $$5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0$$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $$5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0$$

A)
$$\begin{array}{l}\theta=45^{\circ} \\y^{2}=-4 x^{\prime}\end{array}$$
parabola
vertex at $(0,0)$
focus at $(-1,0)$

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

C)
$\begin{array}{l}\theta=45^{\circ} \\x^{\prime 2}=-4 y^{\prime} \\\text { parabola } \\\text { vertex at }(0,0) \\\text { focus at }(0,-1) \end{array}$

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