Rotate the Axes So That the New Equation Contains No $24 x y - 7 y ^ { 2 } + 36 = 0$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equati
- $24 x y - 7 y ^ { 2 } + 36 = 0$

A) $\theta = 36.9 ^ { \circ }$
$\frac { y ^ { \prime 2 } } { 9 } - \frac { x ^ { \prime 2 } } { 16 } = 1$
hyperbola
$\text { center at }(0,0)$
transverse axis is the $y ^ { \prime }$ -axis
$\text { vertices at }(0, \pm 3)$


B) $\theta = 36.9 ^ { \circ }$
$\frac { 4 y ^ { \prime 2 } } { 9 } - \frac { x ^ { \prime 2 } } { 4 } = 1$
hyperbola
$\text { center at }(0,0)$
transverse axis is the $y ^ { \prime }$ -axis
vertices at $\left( 0 , \pm \frac { 3 } { 2 } \right)$


C) $\theta = 36.9 ^ { \circ }$
$\frac { y ^ { \prime 2 } } { 4 } - \frac { 4 x ^ { \prime 2 } } { 9 } = 1$
hyperbola
center at $( 0,0 )$
transverse axis is the $y ^ { \prime }$ -axis
vertices at $( 0 , \pm 2 )$

D) $\theta = 53.1 ^ { \circ }$
$\frac { y ^ { \prime 2 } } { 4 } - \frac { 4 x ^ { \prime 2 } } { 9 } = 1$
hyperbola
center at $( 0,0 )$
transverse axis is the $y ^ { \prime }$ -axis
vertices at $( 0 , \pm 2 )$