Rotate the Axes So That the New Equation Contains No $17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$

Rotate the axes so that the new equation contains no xy-term. Discuss the new equation.
- $17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$

A) $\theta = 26.6 ^ { \circ }$
$$\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 16 } = 1$$
ellipse
center at $( 0,0 )$
major axis is $y ^ { \prime } -$ axis
vertices at $( 0 , \pm 4 )$
B) $\theta = 63.4 ^ { \circ }$
$$\frac { \left( x ^ { \prime } - \frac { 2 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } + \frac { 4 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 4 } = 1$$
ellipse
center at $\left( \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)$
major axis is $x ^ { \prime }$ -axis
vertices at $\left( 4 + \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)$ and $\left( - 4 + \frac { 2 \sqrt { 5 } } { 5 } , - \frac { 4 \sqrt { 5 } } { 5 } \right)$
C) $\theta = 63.4 ^ { \circ }$
$x ^ { \prime 2 } = - 16 y ^ { \prime }$
parabola
vertex at $( 0,0 )$
focus at $( 0 , - 4 )$
D) $\theta = 63.4 ^ { \circ }$
$$\frac { x ^ { \prime 2 } } { 16 } - \frac { y ^ { \prime 2 } } { 4 } = 1$$
hyperbola
center at $( 0,0 )$
transverse axis is the $x ^ { \prime }$ -axis
vertices at $( \pm 4,0 )$