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 that eliminates the xy-term for a quadratic equation of the form $x^2 + 2xy + y^2$ involves a 45-degree rotation, which corresponds to using $\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$. This matches the transformation formulas in option D.

Determine the Appropriate Rotation Formulas to Use So That the New