Rotate the Axes So That the New Equation Contains No $\mathrm { xy } + 16 = 0$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equati
- $\mathrm { xy } + 16 = 0$

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 $y ^ { \prime }$ -axis
vertices at $( 0 , \pm 4 \sqrt { 2 } )$

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

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