Determine the appropriate rotation formulas to use so that the new equation contains no xy-term.
-$$x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$$
A) $x = \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 } x ^ { \prime } - \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 } y ^ { \prime }$ and $y = \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 } x ^ { \prime } + \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 } y ^ { \prime }$
B) $x = - y ^ { \prime }$ and $y = x ^ { \prime }$
C) $x = \frac { 1 } { 2 } x ^ { \prime } - \frac { \sqrt { 3 } } { 2 } y ^ { \prime }$ and $y = \frac { \sqrt { 3 } } { 2 } x ^ { \prime } + \frac { 1 } { 2 } y ^ { \prime }$
D) $x = \frac { \sqrt { 2 } } { 2 } \left( x ^ { \prime } - y ^ { \prime } \right)$ and $y = \frac { \sqrt { 2 } } { 2 } \left( x ^ { \prime } + y ^ { \prime } \right)$

Question

Quizplus · Accepted Answer

The rotation formula to eliminate the xy-term for an equation of the form $x^2 + 2xy + y^2$ involves a 45-degree rotation, which corresponds to $\theta = 45^\circ$ or $\theta = \pi/4$ radians. The formulas for $x$ and $y$ in terms of $x'$ and $y'$ using a rotation of $\theta = \pi/4$ are exactly those given in option D, utilizing $\cos(\pi/4) = \sin(\pi/4) = \sqrt{2}/2$.

Determine the Appropriate Rotation Formulas to Use So That the New