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 equati
- $$5 x ^ { 2 } - 6 x y + 5 y ^ { 2 } - 8 = 0$$

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

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